Elliptic Yang-Mills Flow Theory

Home , Morse homology

arXiv:1303.1401v2 [math.DG] 5 Dec 2013 pedxB ute uiir eut 39 28 32 21 30 results density auxiliary energy Further References the for inequalities B. Differential Appendix A. flow Appendix Restricted 5 5 manifolds homology product 10. Morse 4 Three-dimensional Yang–Mills parabolic to 9. Relationship homology Yang–Mills 8. Elliptic theorem 7. Transversality equation point critical 6.3. Perturbed perturbations 6.2. Holonomy 6.1. Transversality decay 6. Exponential 5. Compactness theorem 4. Fredholm spaces operator 3.3. moduli Linearized and lines flow 3.2. theory Gradient Fredholm and spaces 3.1. Moduli equations flow Yang–Mills 3. Elliptic 2. Acknowledgements Introduction 1. 2010 Date eiae oPoesrDemrA aao nteocso of occasion the on Salamon A. Dietmar Professor to Dedicated eray1,2018. 19, February : ahmtc ujc Classification. Subject Mathematics onsof points hoyi ie otredmninlpoutmanifolds product three-dimensional o to application given par an is Finally, well-developed theory surfaces. the Riemannian to of relation homology its discuss furthermore nanwydfie ag-nain functional gauge-invariant defined newly a on rmi h aeo opc w-iesoa aemanifold base two-dimensional compact o a details of p analytical case a complete the via the in cont functional out Yang–Mills gram This carry the We equations. of flow. study elliptic gradient a of to system approaches novel ous a to rise gives principal a on connections Abstract. LITCYN–IL LWTHEORY FLOW YANG–MILLS ELLIPTIC J orsodt agMlscnetoson connections Yang–Mills to correspond elytefudtoso os oooyo h pc of space the on homology Morse a of foundations the lay We R M ANRADJNSWOBODA JAN AND JANNER EMI ´ G bnl vracmatmanifold compact a over -bundle Contents 1 81,5C7 56,53D20. 35J60, 53C07, 58E15, J hl h critical the While . P its , Y Σ = blcMorse abolic i 0hbirthday 60th his u elliptic our f L at previ- rasts 2 u pro- our f -gradient Y × arabolic Y based , S We . 1 . 41 37 34 22 20 20 16 13 7 6 4 2 2 R.JANNERANDJ.SWOBODA

1. Introduction During the last decades many authors, in order to understand Morse theoretical properties of the Yang–Mills functional 1 YM: A(P ) → R, YM(A)= hF ∧ ∗F i 2 A A ZY on a principal G-bundle P over a compact manifold Y , studied its L2- gradient flow ∗ (1) ∂sA + dAFA = 0. This approach was introduced by Atiyah and Bott (cf. [3]) and used for ex- ample by Donaldson (cf. [5]) to prove a generalized version of the Narasimhan– Seshadri theorem. Analytical properties of the solutions of (1) for principal bundles over 2- and 3-dimensional base manifolds, or over base manifolds with a symmetry of codimension 3, were proven by R˚ade (cf. [15, 16]) and Davis (cf. [4]). A consideration common to all of these works is the following one. Since Eq. (1) is invariant under an infinite-dimensional group of symmetries, the group G(P ) of gauge transformations, it is only degenerate parabolic, in contrary to e.g. the heat flow on manifolds. This problem can be overcome by imposing a suitable gauge-fixing condition. The linearized flow equation then splits into an operator of mixed parabolic and elliptic type (cf. [4, 12, 23]). Short-time existence of the thus augmented equation is shown in [15]. Long-time existence holds below the critical dimension n = 4, and is unknown to be case for n = 4 (cf. Struwe’s article [22] for a blow-up analysis in this case). Moreover, it is not known whether the flow satisfies the Morse–Smale transversality property and it is therefore in general an open question whether a Morse homology, based on the Yang–Mills functional and its L2-flow, can be defined in dimensions n ≥ 3.

In this context we present a novel approach to Yang–Mills theory and deﬁne a new Morse homology as follows. Let Y be a closed manifold of dimension n ≥ 2. Fix a Riemannian metric g on Y and let dvolY denote its volume form. We consider a principal G-bundle P → Y , G a compact Lie group with Lie algebra g, and an ad-invariant inner product on g. Let X := A(P ) × Ωn−2(Y, ad(P )) be the product of the space of g-valued connections on P and ad(P )-valued (n−2)-forms. Then we consider the energy functional 1 J : X → R, J (A, ω) := hF , ∗ωi− |ω|2 dvol . A 2 Y ZY Critical points (A, ω) of this functional satisfy the system of equations

∗FA − ω = 0, dAω = 0 ∗ in which case A is a Yang–Mills connection (i.e., satisfies dAFA = 0) and ∗ω is its curvature. The most interesting aspect are the L2-gradient flow ELLIPTIC YANG–MILLS FLOW THEORY 3 equations n+1 (2) ∂sA + (−1) ∗ dAω = 0, ∂sω + ∗FA − ω = 0, a system of first order nonlinear PDEs, which for n = 2 define a nice elliptic problem. Based on the gradient flow equations (2), we define a new elliptic Yang–Mills Morse homology. To make this idea precise, we need to work with a perturbed version J + hf of the functional J . The reason for adding a perturbation hf is twofold. First, it is in general not the case that critical points of the unperturbed functional J are nondegenerate in the Morse–Bott sense. By this we mean that the nullspace of the Hessian H(A,ω)J of J at each critical point (A, ω) consists entirely of infinitesimal gauge transformations. The second reason is that a priori it is not clear whether gradient flow lines of J , connecting two critical points, satisfy Morse–Smale transversality. This condition, which is equivalent to the linearized gradient flow equation at such a flow line being surjective, is required to obtain smooth moduli spaces of solutions. We here work with holonomy perturbations (cf. §6.1 for details) the construction of whose is based on work by Salamon and Wehrheim in [21]. The precise form of the 2 L -gradient flow equations resulting from the perturbed functional J + hf will be stated in (3) below.

The plan of the article is as follows. In §2 we introduce the elliptic Yang– Mills equations. We then move on to establish their main analytical properties as far as they are needed to define elliptic Yang–Mills homology in the case n = 2 (which we introduce in Section 7). We in particular show (cf. Section 3) that the linearization of the gradient flow equations give rise to a Fredholm operator and determine its index. The main difficulty one encounters here is that the functional J is neither bounded from below nor from above so that the number of eigenvalue crossings of the resulting spectral flow cannot be read off from the index of the Hessian at limiting critical points of J . To overcome this problem we relate this spectral flow to that of a further family of elliptic operators, the numbers of negative eigenvalues of their limits as s → ±∞ being finite and equal to the index of the limiting Yang–Mills connections. A further result concerns compactness up to gauge transformations and convergence to broken trajectories of the moduli spaces of solutions to (3) which have uniformly bounded energy, cf. §4. Exponential decay of finite energy solutions of (3) towards critical points is shown in §5. Transversality is being dealt with in §6. A Morse boundary operator defined by counting the numbers of elements in suitable moduli spaces of connecting flow lines is introduced in §7. Together with the critical points of J + hf in a fixed sublevel set it gives rise to a chain complex, the homology of which we name elliptic Yang–Mills Morse homology.

In §8 we initiate a comparison of elliptic Yang–Mills Morse homology with its classical parabolic counterpart (cf. [3, 23]). Although these theories are 4 R.JANNERANDJ.SWOBODA based on very different types of equations, they share a number of common properties. Namely, the set of generators of the relevant chain complexes is the same in both cases, and also the dimensions of moduli spaces of connecting flow lines agree in each case. Therefore it seems natural to conjecture that there exists an isomorphism between elliptic and parabolic Morse homology. We give some evidence to this conjecture by constructing a so-called hybrid moduli space in which we combine elliptic and parabolic flow lines. Leaving the analytical details of that construction to a future publication we anticipate, based on a crucial energy inequality, that arguments similar to those in [1, 2, 24] apply and give rise to an invertible chain homomorphism between the parabolic and the elliptic Morse complex.

As an application of the elliptic Yang–Mills homology presented here we consider in §9 three-dimensional product manifolds Y =Σ × S1. We relate the new invariants obtained in this case to various other homology groups, amongst them Floer homology of the cotangent bundle of the space of gauge equivalence classes of ﬂat SO(3) connections over Σ.

One difficulty in extending elliptic Morse homology to manifolds Y of dimension n ≥ 3 consists in the fact that the linearization of equation (2) ceases to be elliptic, even if appropriate gauge-fixing conditions are imposed. To overcome this problem we introduce in §10 a modification of the above setup. The main idea is to restrict the configuration space X = A(P ) × Ωn−2(Y, ad(P )) to the Banach submanifold ∗ X1 := {(A, ω) ∈ X | dAω = 0} of X and consider the flow (2) on X1 instead. This modification is natural because the critical points of J are automatically contained in X1. As it turns out, the linearization obtained through this modification are in fact elliptic equations, however of nonlocal type. A discussion of their compactness and transversality properties is left to future work. Acknowledgements. The second named author gratefully acknowledges DFG for its financial support through grant SW 161/1-1. He also thanks the Department of Mathematics of Stanford University for its hospitality where part of this work has been carried out.

2. Elliptic Yang–Mills flow equations Let Y be a closed oriented smooth manifold of dimension n ≥ 2, endowed with a Riemannian metric g. We keep the notation as introduced before. For pairs (A, ω) ∈ X = A(P ) × Ωn−2(Y, ad(P )) we consider the functional 1 J : X → R, J (A, ω)= hF , ∗ωi− |ω|2 dvol . A 2 Y ZY For transversality reasons apparent later we add a so-called holonomy per- 2 turbation hf to J which we shall define in §6.1. The L -gradient of J + hf ELLIPTIC YANG–MILLS FLOW THEORY 5 is given by n+1 ∇J (A, ω)= (−1) ∗ dAω − Xf (A, ω), ∗FA − ω + Yf (A, ω) . 2 For a definition of the term (Xf (A, ω),Yf (A, ω)) which results as the L - gradient of hf , cf. (33) below. We define the perturbed elliptic Yang– Mills flow to be the system of equations 0= ∂ A − d Ψ + (−1)n+1 ∗ d ω − X (A, ω), (3) s A A f (0= ∂sω + [Ψ,ω] − ω + ∗FA + Yf (A, ω). for a connection A ∈ A(P ) and ad(P )-valued forms Ψ ∈ Ω0(Y, ad(P )) and ω ∈ Ωn−2(Y, ad(P )). The term Ψ has been introduced in (3) in order to make this system of equations invariant under the action of time-dependent gauge transformations. Stationary points of the unperturbed flow equation (referring to the unperturbed functional J ) such that Ψ = 0 satisfy

(4) dAω =0 and ω = ∗FA.

Then dAω = dA ∗ FA = 0 and hence the set of critical points of J is in bijection with the set of Yang–Mills connections on the bundle P . Remark 1. The factor (−1)n+1 appearing in the first equation in (3) results ∗ n(deg ω+1)+1 from the different signs the formal adjoint dA = (−1) ∗ dA ∗ ω of dA has in different degrees and dimensions. Remark 2. Assume (A, ω) is a solution of the unperturbed equation (3) on I × Y , I a bounded or unbounded interval. Assume in addition that Ψ = 0. Then it follows that ω satisfies the linear PDE ∗ (5) 0 =ω ¨ − ω˙ − dAdAω. In the case n = 2 this is an elliptic equation with the Hodge Laplacian d2 0 − ds2 +∆A acting on Ω (I × Y, ad(P )) as leading order term. 3. Moduli spaces and Fredholm theory From now on we specialize to the case n = 2 and let Y = Σ denote a closed oriented surface, being endowed with a Riemannian metric g. 3.1. Gradient flow lines and moduli spaces. Throughout we call a smooth solution (A, ω, Ψ) of (3) on R × Σ a negative gradient flow line of J . We say that (A, ω, Ψ) is in temporal gauge if Ψ = 0. The energy of a solution (A, ω, Ψ) of (3) is

Ef (A, ω, Ψ) :=

1 2 2 k∗dAω+dAΨ+Xf (A, ω)kL2(Σ)+k∗FA−ω+[Ψ∧ω]+Yf (A, ω)kL2(Σ) ds. 2 R Z It is by definition invariant under time-dependent gauge transformations. In §5 it will be shown that a temporally gauged solution (A, ω, 0) of (3) on R×Σ ± ± has finite energy if and only if there exist critical points (A ,ω ) of J + hf 6 R.JANNERANDJ.SWOBODA such that (A(s),ω(s)) converges exponentially to (A±,ω±) as s → ±∞. − − + + We let Mf (A ,ω , A ,ω ) denote the moduli space of gauge equivalence classes of negative gradient flow lines from (A−,ω−) to (A+,ω+), i.e. M (A−,ω−, A+,ω+) M (A−,ω−, A+,ω+) := f , f G(P ) c where (A, ω, 0) satisfies (3), E (A, ω, 0) < ∞, M (A−,ω−, A+,ω+) := (A, ω) f . f lim (A(s),ω(s)) ∈ [(A±,ω±)] s→±∞

Thec moduli space M (A−,ω−, A+ ,ω+) can be described in a slightly differ- f ent way as the quotient of all finite energy gradient flow lines (A, ω, Ψ) from (A−,ω−) to (A+,ω+) such that Ψ(s) → 0 as s → ±∞ modulo the action of the group G(R×Σ) of smooth time-dependent gauge transformations, which converge exponentially to the identity as s → ±∞. One of our goals in the − − + + subsequent sections is to show that the moduli space Mf (A ,ω , A ,ω ) is a compact manifold with boundary and to determine its dimension. We start by considering first the linearized operator for Eq. (3).

3.2. Linearized operator. Linearizing the map ∂ A − d Ψ − ∗d ω − X (A, ω) F : (A, ω, Ψ) 7→ s A A f ∂ ω + [Ψ,ω] − ω + ∗F + Y (A, ω) s A f at (A, ω, Ψ) yields the linear operator

(6) D(A,ω,Ψ) : (α, v, ψ) 7→

α ∗[ω ∧ · ] − ∗ dA −dA α − dXf (A, ω)(α, v)

∇s v + ∗dA −½ −[ω ∧ · ] v + dYf (A, ω)(α, v) ,    ∗      ψ −dA [ω ∧ · ] 0 ψ 0         =:B(A,ω,Ψ) where ∇ :=| ∂ + [Ψ ∧ · ]. To{z specify its domain} and target we deﬁne for s ∂s p> 1 Lp := Lp(R,Lp(Σ)) and Wp := W 1,p(R,Lp(Σ)) ∩ Lp(R,W 1,p(Σ)), where for abbreviation we set

Lp(R,Lp(Σ)) := Lp(R,Lp(Σ, T ∗Σ ⊗ ad(P ))) ⊕ Lp(R,Lp(Σ, ad(P ))) ⊕ Lp(R,Lp(Σ, ad(P ))), and similarly for Lp(R,W 1,p(Σ)) and W 1,p(R,Lp(Σ)). Throughout we shall p p consider D(A,ω,Ψ) as an operator D(A,ω,Ψ) : W →L . Remark 3 (Gauge-fixing condition). The last component in the definition (6) of the operator D(A,ω,Ψ) is a gauge-fixing condition. It can be understood as follows. We identify the pair (A, Ψ) with the connection A +Ψ ds on the ELLIPTIC YANG–MILLS FLOW THEORY 7 principal G-bundle R × P . The group G(R × P ) of time-dependent gauge transformations acts on pairs (A +Ψ ds,ω) as g∗(A +Ψ ds,ω) = (g∗A + (g−1Ψg + g−1g˙) ∧ ds, g−1ωg). The infinitesimal action at (A +Ψ ds,ω) is the map

ϕ 7→ (dAϕ + (ϕ ˙ + [Ψ ∧ ϕ]) ∧ ds, [ω ∧ ϕ]). A short calculation now shows that (α+ψ ∧ds, v) is orthogonal to the image of the inﬁnitesimal action precisely if the last component of D(A,ω,Ψ)(α, v, ψ) vanishes. 3.3. Fredholm theorem. Throughout this section we ﬁx a > 0 and an a-regular perturbation hf . The aim of this section is to prove the following theorem. Theorem 4. Let (A, ω, Ψ) be a solution of (3) and assume that there exist solutions (A±,ω±) of the perturbed critical point equation (34) such that lim (A(s),ω(s), Ψ(s)) = (A±,ω±, 0) s→±∞ k in C (Σ) for every k ∈ N0. Then for every 1

(7) ind D(A,ω,Ψ) = ind HA−,f − ind HA+,f .

Here we denote by ind HA,f the Morse index (i.e. the number of negative eigenvalues) of the perturbed Yang–Mills Hessian ∗ HA,f := dAdA + ∗[∗FA ∧ · ]+dXf (A). Because the statement of this theorem is invariant under gauge transformations we may from now on assume that Ψ = 0. For ease of notation we also deﬁne Bf := B(A,ω,0) + (− dXf (A, ω), dYf (A, ω)).

Fredholm property and index in the case p = 2. We let p = 2. Then Bf (s) is a self-adjoint operator on the Hilbert space H := L2(Σ) with domain 1,2 W := dom Bf (s) = W (Σ), for every s ∈ R. This follows by the same arguments as given in the proof of Proposition 22. Proof. [Theorem 4 in the case p = 2]. To show the Fredholm property we employ the result [17, Theorem A]. To apply this theorem we need to check that the following conditions (i-iv) are satisfied. (i) The inclusion W ֒→ H of Hilbert spaces is compact with dense range. This holds true by the Rellich–Kontrachov compactness theorem. (ii) The norm of W is equivalent to the graph norm of Bf (s): W → H for every s ∈ R. This follows from a standard elliptic estimate for the operator Bf (s): W → H. (iii) The map R →L(W, H): s 7→ Bf (s) is continuously differentiable with respect to the weak operator topology. For this we need to verify that for every ξ ∈ W and 1 η ∈ H the map s 7→ hBf (s)ξ, ηi is of class C (R). Because s 7→ (A(s),ω(s)) is a smooth path in X and Bf depends smoothly on (A, ω), this property is ± clearly satisfied. (iv) The operators B(A±,ω±,0) −(dXf (A ), 0) are invertible 8 R.JANNERANDJ.SWOBODA and are the limits of Bf (s) in the norm topology as s → ±∞. Invertibility follows because the perturbation hf was assumed to be a-regular. Theorem 14 (exponential decay) gives uniform convergence with all derivatives of (A(s),ω(s)) to (A±,ω±) as s → ±∞, hence in particular norm convergence ± Bf (s) → B(A±,ω±,0) − (dXf (A ), 0) as s → ±∞. We have thus verified that the operator family s 7→ Bf (s) satisfies all the assumptions of [17, Theorem d A]. It hence follows that the operator D(A,ω,0) = ds + Bf (s) is Fredholm with index given by the spectral flow of the family s 7→ Bf (s), cf. below. It is shown in Proposition 5 and Lemma 6 below that this spectral flow equals that of a further operator family s 7→ Cf (s). Its spectral flow in turn is equal to the right-hand side of (7), which is the content of the subsequent Lemma 9. The index formula (7) now follows.

It remains to determine the index of the Fredholm operator D(A,ω,0). This index is related to the spectral ﬂow of the operator family s 7→ Bf (s) as we explain next. Here we follow the discussion in [17, §4]. Recall that a crossing of Bf is a number s ∈ R for which Bf (s) is not injective. The crossing operator at s ∈ R is the map

Γ(Bf ,s): ker Bf (s) → ker Bf (s), Γ(Bf ,s)= P (s)B˙ f (s), where P (s): H → H denotes the orthogonal projection onto ker Bf (s). A crossing s ∈ R is called regular if Γ(Bf (s)) is nonsingular. It can be shown (by adapting the arguments in [17, Theorem 4.2]) that, after replacing f by a suitable, a-regular perturbation fˆ (which may be chosen arbitrarily close to f), the operator family s 7→ Bfˆ(s) only has regular crossings. We assume that f has been chosen accordingly. Then the number of crossings of Bf is in particular ﬁnite. The signature of the crossing s ∈ R is the signature (the number of positive minus the number of negative eigenvalues) of the endomorphism Γ(Bf ,s). The Fredholm index of D(A,ω,0) is determined by the crossing signatures via the relation

(8) ind D(A,ω,0) = − sign Γ(Bf ,s) s X where the sum is over all crossings, cf. [17, Theorem 4.1].

We now aim to determine the Fredholm index of D(A,ω,0) from (8). The following proposition simpliﬁes the subsequent computations.

Proposition 5. Let Bf,0 denote the family of operators which we obtain d 2 2 from Bf by setting Y = 0. Then the operator ds + Bf,0 : W → L is a Fredholm operator with index equal to that of D(A,ω,0). d Proof. For 0 ≤ τ ≤ 1 consider the family of operators ds + B(A,ω,0) + (− dXf (A, ω),τ dYf (A, ω)). Since Yf (A(s),ω(s)) = 0 for suﬃciently large |s| ≥ s0 (which holds by our choice of the a-regular perturbation f) it follows that for each τ the limiting operator as s → ±∞ of B(A,ω,0) + ELLIPTIC YANG–MILLS FLOW THEORY 9

(− dXf (A, ω),τ dYf (A, ω)) is equal to the invertible operator B(A±,ω±,0) − ± (dXf (A , 0). By the same proof as of Theorem 4 in the case p = 2 this implies that each of the operators in the above family is Fredholm. Be- cause this family is an interpolating family of Fredholm operators between d ds + Bf,0 and D(A,ω,0) equality of the indices follows.

To determine the Fredholm index of D(A,ω,0) we use Proposition 5 together with relation (8) (with the operator Bf replaced by Bf,0). The crossing indices of the operator family Bf,0 are found by relating them to those of a further path of operators Cf (s) which we introduce next. For λ ∈ R \ {−1} and (A, ω) = (A(s),ω(s)) we deﬁne the symmetric operator Cf,λ(s) by 1 d∗ d + ∗[ω ∧ · ]+dX (A, ω) −d + 1 ∗ d [ω ∧ · ] C (s) := λ+1 A A f A λ+1 A . f,λ −d∗ + 1 ∗ [ω ∧ d · ] − 1 ∗ [ω ∧ ∗[ω ∧ · ]] A λ+1 A λ+1 We furthermore set Cf (s) := Cf,0(s).

Lemma 6. For every s ∈ R the crossing indices Γ(Bf,0(s)) and Γ(Cf (s)) coincide.

Proof. We ﬁrst note that λ = 0 is an eigenvalue of Bf,0(s) if and only if it is an eigenvalue of Cf (s). In this case, the corresponding eigenspaces are of the same dimension. These facts follow from Proposition 7 below. It remains to show that for every crossing s0 ∈ R the signatures sign Γ(Bf,0,s0) and sign Γ(Cf ,s0) coincide. We prove this equality for a simple crossing s0, i.e. in the case where the crossing is regular and in addition the kernels of Bf,0(s0) and Cf (s0) are one-dimensional. In this situation let s 7→ λ(s), 1 s ∈ (s0 −ε, s0 +ε), be a C -path of eigenvalues of Bf,0(s) with corresponding path of eigenvectors ξ(s) = (α(s), v(s), ψ(s)) such that λ(s0) = 0. Similarly, 1 let s 7→ µ(s), s ∈ (s0 − ε, s0 + ε), be a C -path of eigenvalues of Cf (s) with normalized eigenvectors ξ˜(s) = (˜α(s), ψ˜(s)) such that µ(s0) = 0 and ξ˜(s0) = (α(s0), ψ(s0)). It suﬃces to prove that

(9) sign λ˙ (s0) = signµ ˙ (s0). To prove this claim we ﬁrst apply Proposition 7 below which gives the identity

Cf,λ(s)(s)(α(s), ψ(s)) = λ(s)(α(s), ψ(s)) for all s ∈ (s0 − ε, s0 + ε). Next, by deﬁnition of the operator families Cf,λ(s)(s) and Cf (s) it follows that (10) ∗ d ˙ ˙ −dAdA − ∗ dA[ω ∧ · ] Cf,λ(s)(s)= Cf (s0)+λ(s0) . ds − ∗ [ω ∧ dA · ] ∗[ω ∧ ∗[ω ∧ · ]] s=s0

We apply Proposition 8 below to the operator family s 7→ C (s). To- f,λ(s) gether with (10) it then follows that (we abbreviate A := A(s0), ω := ω(s0) 10 R.JANNERANDJ.SWOBODA and recall that ξ˜(s0) = (α(s0), ψ(s0))) d λ˙ (s )= C (s)ξ˜(s ), ξ˜(s ) 0 ds f,λ(s) 0 0 s=s0 D E =hC˙ (s )ξ˜(s ), ξ˜(s )i f 0 0 0 −d∗ d − ∗ d [ω ∧ · ] + λ˙ (s ) A A A ξ˜(s ), ξ˜(s ) 0 − ∗ [ω ∧ d · ] ∗[ω ∧ ∗[ω ∧ · ]] 0 0 A ˙ ∗ ˙ =µ ˙ (s0) − λ(s0)hdAdAα(s0), α(s0)i + 2λ(s0)h∗dAα(s0), [ω ∧ ψ(s0)]i

+ λ˙ (s0)h∗[ω ∧ ∗[ω ∧ ψ(s0)], ψ(s0)i 2 2 =µ ˙ (s0) − λ˙ (s0)kdAα(s0)k − λ˙ (s0)k[ω ∧ ψ(s0)]k

+ 2λ˙ (s0)h∗dAα(s0), [ω ∧ ψ(s0)]i.

Here we used again Proposition 8 in order to replace hC˙ f (s0)ξ˜(s0), ξ˜(s0)i = µ˙ (s0). Concerning the remaining terms in the same line we compute

hα(s0), − ∗ dA[ω ∧ ψ(s0)]i =h∗α(s0), dA[ω ∧ ψ(s0)]i ∗ =hdA ∗ α(s0), [ω ∧ ψ(s0)]i =h∗dAα(s0), [ω ∧ ψ(s0)]i, and likewise for h− ∗ [ω ∧ dAα(s0)], ψ(s0)i. It ﬁnally follows that 2 2 µ˙ (s0)= 1+kdAα(s0)k +k[ω∧ψ(s0)]k −2h∗dAα(s0), [ω∧ψ(s0)]i λ˙ (s0). Using the Cauchy–Schwartz inequality we can estimate the factor in front of λ(s0) in the previous line as 2 2 1+ kdAα(s0)k + k[ω ∧ ψ(s0)]k − 2h∗dAα(s0), [ω ∧ ψ(s0)]i 2 2 ≥1+ kdAα(s0)k + k[ω ∧ ψ(s0)]k − 2kdAα(s0)kk[ω ∧ ψ(s0)]k ≥1.

Therefore,µ ˙ (s0) and λ˙ (s0) have the same sign which shows (9) in the case of a simple crossing s0. The case of a general regular crossing can be treated similarly. This completes the proof.

Proposition 7. Let the operators Bf,0(s) and Cf,λ(s), s ∈ R, be as defined above. Let λ ∈ R \ {−1}. Then ξ = (α, v, ψ) ∈ dom Bf,0(s) satisfies the eigenvalue equation Bf,0(s)ξ = λξ if and only if ξ˜ = (α, ψ) satisfies the nonlinear eigenvalue equation

(11) Cf,λ(s)ξ˜ = λξ.˜

Proof. Assume ξ = (α, v, ψ) ∈ dom Bf,0(s) satisﬁes

(12) Bf,0(s)ξ = λξ for some λ 6= −1. Then it follows from the deﬁnition of Bf,0(s) that 1 (13) v = (∗d α − [ω ∧ ψ]). λ + 1 A ELLIPTIC YANG–MILLS FLOW THEORY 11

Inserting this v into the first and last of the three equations in (12) yields (11). Conversely, assume that ξ˜ = (α, ψ) satisfies (11) for some λ 6= −1. Defining v by equation (13) and setting ξ := (α, v, ψ) we obtain a solution ξ of (12) for this eigenvalue λ. 1 Proposition 8. Let s 7→ F (s), s ∈ (s0 − ε, s0 + ε), be a C -path of densely defined symmetric operators on a Hilbert space H. Let s 7→ λ(s), 1 s ∈ (s0 −ε, s0 +ε), be a C -path of eigenvalues of the operator family F with normalized eigenvectors x(s). Then it follows that

λ˙ (s0)= hF˙ (s0)x(s0),x(s0)i.

Proof. We diﬀerentiate the eigenvalue equation F (s)x(s) = λ(s)x(s) at s0 and obtain

F˙ (s0)x(s0)+ F (s0)x ˙(s0)= λ˙ (s0)x(s0)+ λ(s0)x ˙(s0).

Take the inner product of both sides with x(s0) and use that by symmetry of F (s0)

hF (s0)x ˙(s0),x(s0)i = hx˙(s0), F (s0)x(s0)i = hx˙(s0), λ(s0)x(s0)i to conclude the result.

Lemma 9. The total number s∈R Γ(Cf (s)) of eigenvalue crossings of the operator family Cf equals the right-hand side of (7). P ± ± ± Proof. We set Cf := lims→±∞ Cf (s). By assumption, each pair (A ,ω )= ± lims→±∞(A(s),ω(s)) satisfies the critical point equations ω = ∗FA± and ± ± ∗dA± ω + Xf (A ) = 0. Inserting the first one into the definition of Cf it follows that

∗ ± ± ± ± ± ± ± dA± dA + ∗[∗FA ∧ · ] − dXf (A ) −dA + ∗dA [∗FA ∧ · ] Cf = ∗ . −d ± + ∗[∗F ± ∧ d ± · ] − ∗ [∗F ± ∧ [F ± ∧ · ]] A A A A A ± We point out that the upper left entry of Cf equals the perturbed Yang– Mills Hessian HA±,f . For 0 ≤ τ ≤ 1 we consider the smooth path

± HA±,f −dA± + τ ∗ dA± [∗FA± ∧ · ] Cτ,f := ∗ : −d ± + τ ∗ [∗F ± ∧ d ± · ] −τ ∗ [∗F ± ∧ [F ± ∧ · ]] A A A A A W 2,2(Σ) ⊕ W 1,2(Σ) → L2(Σ) ⊕ L2(Σ). ± of bounded symmetric operators. Note that C0,f equals the perturbed Yang– ± Mills Hessian HA±,f , augmented by a gauge-ﬁxing condition, while C1,f = ± ± Cf . We claim that 0 is not an eigenvalue of Cτ,f , for every 0 ≤ τ ≤ 1. To ± prove this claim, assume that (α, ψ) satisﬁes Cτ,f (α, ψ) = 0. Then it follows that

HA±,f α − dA± ψ + τ ∗ dA± [∗FA± ∧ ψ] = 0. ∗ Applying dA± to both sides of the last equation yields ∗ ∗ dA± dA± ψ − τdA± ∗ dA± [∗FA± ∧ ψ] = 0. 12 R.JANNERANDJ.SWOBODA

± ∗ 2 This follows because im HA ,f ⊆ ker dA± . Take the L -inner product with ψ to obtain ∗ ∗ 0= hψ, dA± dA± ψ − τdA± ∗ dA± [∗FA± ∧ ψ]i = hdA± ψ, dA± ψi− τhψ, ∗dA± dA± [∗FA± ∧ ψ]i

= hdA± ψ, dA± ψi− τhψ, ∗[FA± ∧ [∗FA± ∧ ψ]i 2 2 = kdA± ψk + τk[FA± ∧ ψ]k . It follows that ψ = 0 because the connections A± are irreducible. We hence ± ∗ ± ± conclude that HA ,f α = 0 and that dA± α − τ ∗ [∗FA ∧ dA α] = 0. Because the perturbed Hessian HA±,f is nondegenerate it follows that α = dA± ϕ for some ϕ ∈ Ω0(Σ, ad(P )). Then the second of these equations implies that ∗ 0= dA± dA± ϕ − τ ∗ [∗FA± ∧ dA± dA± ϕ] ∗ = dA± dA± ϕ − τ ∗ [∗FA± ∧ [FA± ∧ ϕ]], from which it follows as before, by taking the inner product with ϕ and using irreducibility of A±, that ϕ = 0 and hence also α = 0. We now ± argue by continuity of the spectral flow of the operator family τ 7→ Cτ,f ± that the (finite) number of negative eigenvalues of Cτ,f does not change with τ. The statement of the lemma now follows because the total number of eigenvalue crossings of the operator family s 7→ Cf , s ∈ R, equals the − + difference of the numbers of negative eigenvalues of Cf and Cf . By the preceding argumentation this difference is equal to the difference between the Morse indices of the augmented Yang–Mills Hessians at A−, respectively A+, whence the first claim.

Fredholm property and index in the case 1 0 smaller than the modulus of any eigenvalue of the limiting opera- ± tors Bf . Such a choice is possible since these operators are by assumption nondegenerate. Step 1. Let N be the vector space ∞ R R N = {ξ = (α, v, ψ) ∈ C ( × Σ) | D(A,ω,0)ξ = 0, ∃c> 0 ∀s ∈ : −δ|s| kξ(s)kL∞(Σ) + k∂sξ(s)kL∞(Σ) + k∇Aξ(s)kL∞(Σ) ≤ ce }. p p Then ker(D(A,ω,0) : W → L ) = N. In particular, this kernel is ﬁnite- dimensional and does not depend on p.

By standard elliptic estimates, any solution of D(A,ω,0)ξ = 0 is smooth. Exponential decay in L2(Σ) of any such ξ holds by (25). Elliptic bootstrapping arguments then show exponential decay in the form stated above. This ELLIPTIC YANG–MILLS FLOW THEORY 13

p p proves the inclusion ker(D(A,ω,0) : W → L ) ⊆ N. The opposite inclusion is clearly satisfied. Step 2. Let N ∗ be the vector space ∗ ∞ R ∗ R N = {η ∈ C ( × Σ) | D(A,ω,0)η = 0, ∃c> 0 ∀s ∈ : −δ|s| kη(s)kL∞(Σ) + k∂sη(s)kL∞(Σ) + k∇Aη(s)kL∞(Σ) ≤ ce }. ∗ p p ∗ Then coker(D(A,ω,0) : W →L )= N . In particular, this cokernel is finite- dimensional and does not depend on p. The statement follows from Step 1 using the reflection s 7→ −s under ∗ which D(A,ω,0) is mapped to −D(A,ω,0). p p Step 3. The range of the operator D(A,ω,0) : W →L is closed.

First, there exists a constant c = c(A,ω,p) > 0 such that for any s0 ∈ R and the two intervals I1 = (s0 − 1,s0 + 1) and I2 = (s0 − 2,s0 + 2) the local elliptic estimate

1,p p p kξkW (I1×Σ) ≤ c(kD(A,ω,0)ξkL (I2×Σ) + kξkL (I2×Σ)) is satisfied for all ξ ∈ W 1,p(I × Σ). Second, for the two s-independent ± ± 1,p ± p ± solutions (A ,ω , 0) of (3) the operator D(A±,ω±,0) : W (Z ) → L (Z ) has a bounded inverse. Here we let Z± denote the half-infinite cylinders Z− = (−∞, 0) × Σ, respectively Z+ = (0, ∞) × Σ. By a standard cut-off function argument, both estimates can be combined into the further estimate

kξkWp ≤ c(kD(A,ω,0)ξkLp + kKξkLp ), where K : Wp → Lp is a suitable compact operator. The assertion now follows from the abstract closed range lemma, cf. [18, p. 14]. Step 4. We prove the theorem. p p By the preceding steps, the operator D(A,ω,0) : W → L is a Fredholm operator, its index being independent of p. Hence the asserted index formula (7) follows from the one already established in the case p = 2.

4. Compactness In this section we prove a compactness theorem for solutions of (3) of uniformly bounded energy. Let I ⊆ R be an interval. For convenience we shall identify at several instances a path (A, Ψ) ∈ C∞(I, A(P ) × Ω0(Σ, ad(P )))

with the connection A := A +Ψ ds ∈A(I × P ). Its curvature is 2

(14) FA = FA + (∂sA + dAΨ) ∧ ds ∈ Ω (I × Σ, ad(I × P )).

We call a connection A1 ∈A(I × P ) to be in local slice with respect to

A A the reference connection A if the diﬀerence 1 − = α + ψ ds satisﬁes

∗ dA (α + ψ ds) = 0. 14 R.JANNERANDJ.SWOBODA

This condition is equivalent to ∗ (15) ∇sψ − dAα = 0, where we denote ∇sψ := ∂sψ + [Ψ, ψ].

0 0

In the following we ﬁx (A0,ω0, Ψ0) ∈ A(P ) × Ω (Σ, ad(P )) × Ω (Σ, ad(P )) A as a smooth reference point and denote A0 := A0 +Ψ0 ds. Thus 0 is a connection on I × Σ whose components do not depend on the time-parameter s. Let (A, ω, Ψ) = (A0,ω0, Ψ0)+ ξ with ξ = (α, v, ψ) be a smooth solution of (3) on I × Σ. We augment (3) with the local slice condition (15) rela- tively to the reference connection A0. Expanding the thus obtained system of equations about (A0,ω0, Ψ0) yields the equation

(16) 0=F(A0,ω0, Ψ0) + (∇s + B(A0,ω0,Ψ0))ξ + Qω0 ξ + (−Xf (A, ω),Yf (A, ω), 0), where the linear operator B(A0,ω0,Ψ0) has been deﬁned in (6) and where we set ˙ A0 − dA0 Ψ0 − ∗dA0 ω0 F(A ,ω , Ψ ) := ω˙ + [Ψ ,ω ] − ω + ∗F 0 0 0  0 0 0 0 A0  0   and α −[α ∧ ψ] − ∗[α ∧ v] Q v := [ψ, v]+ 1 ∗ [α ∧ α] . ω0    2  ψ − ∗ [ω0 ∧ ∗v] We furthermore deﬁne the gauge-invariant energy density of the solution (A, ω, Ψ) of (3) to be the function 1 (17) e(A, ω, Ψ) := (| ∗ d ω + d Ψ+ X (A, ω)|2 2 A A f 2 + | ∗ FA − ω + [Ψ,ω]+ Yf (A, ω)| ): I × Σ → R. Proposition 10. Let I ⊆ R be a compact interval and (Aν,ων , 0) be a sequence of smooth solutions of Eq. (3) on I × Σ in temporal gauge. Assume that there exists a constant C > 0 such that for eν := e(Aν ,ων, 0) ν (18) ke kL1(I×Σ) ≤ C for all ν ∈ N. Then there exists a subsequence, still denoted by (Aν ,ων, 0), and a sequence of gauge transformations gν ∈ G(I×P ) such that the sequence ν ∗ ν ν k (g ) (A ,ω , 0) converges in the C topology, for every k ∈ Æ0. Proof. Combining (3) and (14) it follows that the curvature of the connection ν ν A := A + 0 ds ∈A(I × P ) is ν ν ν ν ν ν FA = FA + (∗dA ω + Xf (A ,ω )) ∧ ds. ELLIPTIC YANG–MILLS FLOW THEORY 15

Assumption (18) together with the deﬁnition (17) of eν implies the uniform curvature bound

ν ν ν ν ν 2 ν 2 2 kFA kL (I×Σ) ≤ k ∗ FA − ω + Yf (A ,ω )kL (I×Σ) + kω kL (I×Σ) ν ν ν ν ν + kYf (A ,ω )kL2(I×Σ) + k ∗ dAν ω + Xf (A ,ω )kL2(I×Σ) ≤ C1 ν for some further constant C1. The term kω kL2(I×Σ) is uniformly bounded as follows from Lemmata 28 and 29. A standard estimate allows to uniformly 2 ν ν bound the L -norm of Yf (A ,ω ), cf. [21, Proposition D.1 (iii)]. Therefore the assumptions of Uhlenbeck’s weak compactness theorem (cf. [27, Theorem A]) are satisﬁed for the Sobolev exponent p = 2. It yields the existence of a sequence of gauge transformations gν ∈ G2,p(I × P ) such that after passing to a subsequence

ν ∗ ν ∗ ∗ ∗ A (g ) A ⇀ = A +Ψ ds (ν →∞) 1,2 ∗ 1,2 weakly in W (I × Σ) for some limiting connection A ∈ W (I × Σ). Let

2,2 ˜ ∗ ∗ A g0 ∈ G (I × Σ) be a gauge transformation such that A0 := g0 is in tem-

˜ ˜ A poral gauge, i.e. of the form A0 = A0 +0 ds. Let 0 = A0 +0 ds be a smooth 1,2 ˜ connection W close to A0. Because the weakly convergent sequence ν ∗ ν 1,2 (g ) A is bounded in W , this is also the case for the gauge transformed ν ∗ ν sequence (g g0) A . Now the local slice theorem (cf. [27, Theorem F]) yields ν ν ν ∗ ν a further sequence h of gauge transformations such that (g g0h ) A is a bounded sequence in W 1,2, which in addition is in local slice with respect

ν ν ν ν ∗ ν

A A to the reference connection A0. Denoting α + ψ ds := (g g0h ) − 0 this means that ∗ ν ν ∗ ν ˙ν

(19) dA0 (α + ψ ds)= dA0 α − ψ = 0 for all ν ∈ N. Differentiation of (3) with respect to s shows that each ων satisfies ν ν ν ν ν ν ν 0= −ω¨ +∆Aν ω +ω ˙ + dAν Xf (A ,ω ) − ∂sYf (A ,ω ). The uniform L2-bound satisfied by ων together with ellipticity of the linear 2 d d ν ν operator − ds2 + ds +∆A shows that the sequence ω is in fact uniformly 2,2 ν ν bounded in W . This uses in addition that the terms dAν Xf (A ,ω ) and ν ν 2 ∂sYf (A ,ω ) are uniformly bounded in L which follows from the estimates on holonomy perturbations in [21, Proposition D.1 (iv,vi)]. We now choose 0 ν ν ω0 ∈ Ω (Σ, ad(P )) for reference and set v := ω − ω0. By (16) and (19) each ξν satisfies the equation ν ν ν ν ν ν (∇s + B(A0,ω0,0))ξ = −F(A0,ω0, 0) − Qω0 ξ + (Xf (A ,ω ), −Yf (A ,ω ), 0). The last three terms on the right-hand side of this equation are uniformly 3 3 1, 2 1,2 1, 2 bounded in W (I ×Σ). Namely, the map Qω0 : W (I ×Σ) → W (I ×Σ) maps bounded sets to bounded sets as follows from Sobolev multiplication and embedding theorems, and ξν satisfies a uniform bound in W 1,2(I × Σ). ν ν ν ν The same holds true for the term (Xf (A ,ω ), −Yf (A ,ω ), 0) by a standard estimate, cf. [21, Proposition D.1 (vi)]. The statement of the proposition 16 R.JANNERANDJ.SWOBODA now follows inductively from a bootstrap argument based on ellipticity of the linear operator ∇s + B(A0,ω0,0). The following theorem states compactness of moduli spaces up to convergence to broken trajectories. The notion of a-regular perturbations is introduced in Definition 15 below. We refer to §6.2 for explanation of some of the subsequently used notations. Theorem 11. Fix a number a > 0 and let f be an a-regular perturbation. ν ν Let A− and A+ be two sequences in Crit(J + hf ) which converge uniformly − + ν ν to some critical point A , respectively A of J + hf . Let (A ,ω , 0) be a ν ν sequence in Mf (A−, A+) with bounded energy ν ν ν ν (20) sup(J (A ,ω , 0) + hf (A ,ω )) < a. c ν Then there is a subsequence, still denoted by (Aν ,ων, 0), finitely many crit- − + a ical points B0 = A ,...,Bℓ = A ∈ Crit (J + hf ) and connecting trajec- ν ν tories (Ai,ωi, Ψi) ∈ Mf (Bi,Bi+1) for i = 0,...,ℓ − 1, such that (A ,ω , 0) converges to the broken trajectory c ((A0,ω0, Ψ0),..., (Aℓ−1,ωℓ−1, Ψℓ−1)) ν in the following sense. For every i = 0,...,ℓ − 1 there is a sequences si ∈ R ν R and a sequence of gauge transformations gi ∈ G( × Σ) such that the ν ∗ ν ν ν ν s-dilated sequence (gi ) (A ( · + si ),ω ( · + si ), 0) converges uniformly to (Ai,ωi, Ψi) on compact subsets of R × Σ. Proof. We first note that our assumptions imply for every compact interval I ⊆ R the existence of a constant C > 0 such that uniform energy bound (18) is satisfied. This follows for I = R (and hence for every subinterval I) from (20) and the identity ∞ ν 2 ke (s)kL2(Σ) ds = −∞ Z ν ν ν ν ν ν ν ν J (A−,ω−, 0) −J (A+,ω+, 0) + hf (A−,ω−) − hf (A+,ω+). Hence the restriction of (Aν,ων , 0) to each compact interval I ⊆ R satisfies the assumptions of Proposition 10. Therefore, after passing to a subsequence ν ν and modification by gauge transformations it follows that (A |I ,ω |I , 0) con- ∗ ∗ ∗ verges uniformly to a limit (A |I ,ω |I , Ψ |I ), which again satisfies Eq. (3) on I × Σ. The remaining parts of the statement now follow from standard arguments, which e.g. can be found in [21, §7].

5. Exponential decay Throughout we ﬁx a constant a> 0.

Proposition 12 (Stability of injectivity). Let hf be an a-regular perturbation in the sense of Deﬁnition 15. Then there are positive constants δ˜ ELLIPTIC YANG–MILLS FLOW THEORY 17 and c such that the following holds. Let (A, ω) ∈A(P ) × Ω0(Σ, ad(P )) with J (A, ω) < a satisfy ˜ (21) k ∗ dAω + Xf (A, ω)kL∞(Σ) + k ∗ FA − ω + Yf (A, ω)kL∞(Σ) ≤ δ. Then for every ξ = (α, v, ψ) of class C1, where α ∈ Ω1(Σ, ad(P )) and v, ψ ∈ Ω0(Σ, ad(P )), it holds that 2 2 2 2 (22) kξkL2(Σ) = kαkL2(Σ) + kvkL2(Σ) + kψkL2(Σ) 2 ≤ c k∗dAv + dAψ − ∗[ω ∧ α]+dXf (A, ω)(α, v)kL2(Σ) 2 ∗ 2 + k∗dAα − v − [ω ∧ ψ]+d Yf (A, ω)(α, v)kL2(Σ) + kdAα − ∗[ω ∧ ∗v]kL2(Σ) . (This estimate shows stable invertibility of the Hessian of the perturbed functional J + hf near critical points, cf. (6)). Proof. We assume by contradiction that there is (A, ω) as in the statement of the proposition but (22) fails to be true for every c> 0. Then there is a 0 sequence (Aν ,ων) ∈ A(P ) × Ω (Σ, ad(P )) and a nullsequence δν → 0 such that ν ν ν ν ∞ ∞ (23) k ∗ dAν ων + Xf (A ,ω )kL (Σ) + k ∗ FAν − ων + Yf (A ,ω )kL (Σ) ≤ δν and inequality (22) does not hold for the constant cν = ν. For any (A, ω) we obtain from the deﬁnition of J and the Cauchy–Schwartz inequality that

1 2 kωk 2 = J (A, ω) − h∗ω ∧ (F − ∗ω)i 2 L (Σ) A ZΣ 1 2 2 ≤ a + kωk 2 + k ∗ ω − F k 2 . 4 L (Σ) A L (Σ)

This inequality together with (21) shows that ων and FAν are uniformly 2 bounded in the L -norm. Again by (21), dAν ων = ∇Aν ων is uniformly ∞ bounded in L . Hence altogether, the sequence ων is uniformly bounded 1,∞ ∞ in W . Furthermore, (21) shows that FAν is uniformly bounded in L and thus by Uhlenbeck’s weak compactness theorem (cf. [27, Theorem A]) we can assume that (after modifying by suitable gauge transformations) the 1,p sequence (Aν) has a weakly convergent subsequence in A (P ) for any ﬁxed 2 0, because the critical points below level a are not degenerate. The existence of such a ﬁnite constant c contradicts our above assumption. The assertion now follows.

Theorem 13. Let hf be an a-regular perturbation in the sense of Deﬁnition 15. Then there exist positive constants δ, δ˜, and c such that the following holds. Let (A, ω, Ψ) satisfy (3) on R × Σ such that J (A(s),ω(s)) < a for all s ∈ R. Assume that there exists s0 > 0 such that (A, ω, Ψ) satisﬁes ˜ (24) k∂sA(s) − dA(s)Ψ(s)kL∞(Σ) + k∇sω(s)kL∞(Σ) ≤ δ 18 R.JANNERANDJ.SWOBODA

2 for all |s| > s0. Then for every ξ = (α, v, ψ) of class C , where α(s) ∈ 1 0 Ω (Σ, ad(P )) and v(s), ψ(s) ∈ Ω (Σ, ad(P )), which satisﬁes D(A,ω,Ψ)ξ = 0 and does not diverge as s → ±∞, it follows that

−δ|s| (25) kξ(s)kL2(Σ) ≤ ce for all s>s0. Proof. For ﬁxed ξ = (α, v, ψ) as in the assumptions let us consider the function

1 2 2 2 f : R → R, s 7→ kα(s)k 2 + kv(s)k 2 + kψ(s)k 2 . 2 L (Σ) L (Σ) L (Σ) We claim that f satisﬁes the diﬀerential inequality

(26) f ′′(s) ≥ δ2f(s) for a constant δ > 0 and all s ≥ 1. Assuming this claim for a moment it implies that f decays exponentially as s → ±∞. Namely, since for s ≥ 1 d e−δs f ′(s)+ δf(s) = e−δs f ′′(s) − δ2f(s) ≥ 0, ds it follows that f ′(s)+δf(s) < 0. Otherwise the function s 7→ e−δs (f ′(s)+ δf(s)) would for some s0 ≥ 1 be nonnegative. Since it is increasing it would then be nonnegative for all s ≥ s0. Thus, since f(s) is by assumption bounded, −δs ′ s 7→ e f(s) would decrease and hence f (s) increase for s ≥ s0 suﬃciently large. Therefore f(s) would be unbounded which is a contradiction and hence f ′(s)+ δf(s) < 0. Therefore, if the function f satisﬁes (26), then

−δs f(s) ≤ c1e for a suitable constant c1 > 0. We now prove (26). Diﬀerentiating f twice with respect to s, we obtain (27) ′′ 2 2 2 2 2 2 f =k∇sαkL2(Σ) + k∇svkL2(Σ) + k∇sψkL2(Σ) + hα, ∇sαi + hv, ∇svi + hψ, ∇sψi

2 2 2 1 ∗ 2 ≥k∇ αk 2 + k∇ vk 2 + k∇ ψk 2 + kd α − [ω ∧ v]k 2 s L (Σ) s L (Σ) s L (Σ) 2 A L (Σ) 1 2 1 2 + k ∗ d α − v − [ω ∧ ψ]+dY k 2 + k ∗ d v + d ψ +dX − ∗[ω ∧ α]k 2 2 A L (Σ) 2 A A L (Σ) ˜ 2 2 2 − cδ kαkL2(Σ) + kvkL2(Σ) + kψkL2(Σ) , where the second equality follows from the computation below. To obtain the desired inequality (26) we apply (22) after choosing δ˜ > 0 still smaller if necessary. For abbreviation we set

dX :=dXf (A, ω)(α, v) and dX˙ = ∇s(dX) − dXf (A, ω)(α, ˙ v˙) ELLIPTIC YANG–MILLS FLOW THEORY 19

(and likewise for dYf (A, ω)(α, v)) in the following calculation. Using at several places the assumption that D(A,ω,Ψ)ξ = 0, we compute 2 2 2 hα, ∇sαi + hv, ∇svi + hψ, ∇sψi =hα, ∇s (− ∗ [ω ∧ α]+ ∗dAv + dAψ +dX)i ∗ + hv, ∇s (− ∗ dAα + v + [ω ∧ ψ] − dY )i + hψ, ∇s (dAα − ∗[ω ∧ ∗v])i =hα, ∗dA∇sv + [∇s, ∗dA]vi + hα, dA∇sψ + [∇s, dA]ψi

− hα, ∗[∇sω ∧ α]+ ∗[ω ∧∇sα]i + hα, ∇s dXi − hv, ∗dA∇sα + [∇s, ∗dA]αi

+ hv, ∇svi + hv, [∇sω ∧ ψ] + [ω ∧∇sψ]i + hv, ∇s dY i ∗ ∗ + hψ, dA∇sα + [∇s, dA]αi − hψ, ∗[∇sω ∧ ∗v]+ ∗[ω ∧ ∗∇sv]i =h∗dAα, ∗dAα − v − [ω ∧ ψ]+dY i + hα, [∇s, ∗dA]vi ∗ ∗ + hdAα, dAα − [ω ∧ v]i + hα, [∇s, dA]ψi − hα, ∗[∇sω ∧ α]+ ∗[ω ∧∇sα]i + hα, ∇s dXi

+ h∗dAv, ∗dAv − ∗[ω ∧ α]+ dAψ +dXi − hv, [∇s, ∗dA]αi

+ hv, ∇svi + hv, [∇sω ∧ ψ] + [ω ∧∇sψ]i + hv, ∇s dY i ∗ + hdAψ, dAψ + ∗dAv +dX − ∗[ω ∧ α]i − hψ, [∇s, dA]αi − hψ, ∗[∇sω ∧ ∗v]+ ∗[ω ∧ ∗∇sv]i 2 2 =k ∗ dAv + dAψ − ∗[ω ∧ α]+dXk + k ∗ dAα − v − [ω ∧ ψ]+dY k ∗ 2 ˙ ˙ + kdAα − [ω ∧ v]k + hα, Xi + hv, Y i ∗ + hα, [∇s, ∗dA]vi − hψ, [∇s, dA]αi + hα, [∇s, dA]ψi − hv, [∇s, ∗dA]αi − hα, ∗[∇sω ∧ α]i − hψ, ∗[∇sω ∧ ∗v]i + hv, [∇sω ∧ ψ]i Note that the terms appearing in the last three lines are bilinear expressions in α, v or ψ, which with the help of (24) can be bounded through ˜ 2 2 2 cδ kαkL2(Σ) + kvkL2(Σ) + kψkL2(Σ) for some universal constant c. This completes the proof of inequality (27). Thus, using Proposition 12 and choosing δ˜ still smaller if necessary we conclude that

′′ 2 2 2 2 2 (28) f (s) ≥ δ kαkL2(Σ) + kvkL2(Σ) + kψkL2(Σ) = δ f(s), and thus the claimed exponential convergence follows. We can now state and prove the main result of this section.

Theorem 14 (Exponential decay). For a constant a > 0, let hf be an a- regular perturbation of J . Let (A, ω, Ψ) be a smooth solution of (3) on R×Σ such that lim sups∈R J (A(s),ω(s), Ψ(s)) < a and suppose that

2 (29) |∂sA(s, z), ∂sω(s, z), ∂sΨ(s, z)| dvolΣ ds < ∞. R Z ZΣ 20 R.JANNERANDJ.SWOBODA

± ± Then there are critical points (A ,ω ) of J +hf such that (A(s),ω(s), Ψ(s)) converges to (A±,ω±, 0) as s → ±∞. Moreover, there are constants δ > 0 and Ck, k ∈ N0, such that ± ± −δ|s| k(A − A(s),ω − ω(s))kCk([s−1,s+1]×Σ) ≤ Cke for every |s|≥ 1 and k ∈ N0. Proof. We ﬁrst verify that (A, ω, Ψ) meets the assumptions of Theorem 13. Let some constant δ˜ be given. Then (29) implies that for any δ > 0 there exists s0 > 0 such that for all |s|≥ s0

1 2 ke(A, ω, Ψ)k 1 = k(∂ A, ∂ ω, ∂ Ψ)k 2 ≤ δ, L ([s−1,s+1]×Σ) 2 s s s L ([s−1,s+1]×Σ) where e(A, ω, Ψ) denotes the gauge-invariant energy density as deﬁned in (17). Lemma 27 now yields the estimate

ke(A(s),ω(s), Ψ(s))kL∞ (Σ) ≤ Cδ, for every |s| ≥ s0 and some constant C > 0 which only depends on Σ. Therefore assumption (24) of Theorem 13 is satisﬁed. We apply this theorem to ξ = (∂sA, ∂sω, ∂sΨ). This yields existence of the integral ±∞ ± ± ± (A ,ω , Ψ ) := (A(s0),ω(s0)) + (∂sA, ∂sω, ∂sΨ) ds Zs0 and exponential convergence in L2 of (A(s),ω(s), Ψ(s)) to (A±,ω±, Ψ±) as s → ±∞. After applying a suitable time-dependent gauge transformation we may assume that Ψ± = 0. The claimed exponential convergence with respect to the Ck norm for any k ≥ 0 may now be obtained by standard bootstrapping arguments.

6. Transversality 6.1. Holonomy perturbations. To achieve transversality we introduce a perturbation scheme based on so-called holonomy perturbations. Such perturbations have been used in similar situations, cf. [6, 10, 21, 25]. We adapt the construction of holonomy perturbations as carried out in [21, Appendix D] slightly in order to make it well suited in the situation at hand.

In the following it will be convenient to identify the pair (A, ω) ∈ A(P ) × 0 Ω (Σ, ad(P )) with the connection A := A + ω dt on the principal G-bundle 1 1 P × S over Σ × S . We point out that both components A and ω of A do not depend on the second coordinate t ∈ S1. We let A˜(P ) ⊆ A(P × S1) denote the set of such connections. The diagonal action of G(P ) on A(P ) × 0 ˜ ∗ ∗ −1 Ω (Σ, ad(P )) induces an action on A(P ) given by g A = g A + g ωg dt. Let D ⊆ C denote the closed unit disk. For an integer m ≥ 1 let Γm denote the set of sequences γ = (γ1, . . . , γm) of orientation preserving embeddings 1 1 γi : S × D → Σ × S . Every γ ∈ Γm gives rise to a map m ρ = (ρ1, . . . , ρm): D × A˜(P ) → G , ELLIPTIC YANG–MILLS FLOW THEORY 21

˜ A where ρi(z, A) is the holonomy of the connection ∈ A(P ) along the loop θ 7→ γi(θ, z). Let Fm denote the space of real-valued smooth functions on D × Gm invariant under the diagonal action by simultaneous conjugation m of G on G . Each pair (γ,f) ∈ Γm × Fm determines a smooth function

hf : A˜(P ) → R by A (30) hf (A) := f(z, ρ(z, )) dz. D Z The integrand is by construction invariant under the action of G(P ) (in 1 fact of the larger group G(P × S )) on the argument A. By the Riesz representation theorem, the diﬀerential dhf (A)(α + v dt) in direction of α + ˜

v dt ∈ TA A(P ) can be written in the form

A A (31) dhf (A)(α + v dt)= h−Xf ( ) ∧ ∗αi + hYf ( ) ∧ ∗vi ZΣ

1 0 A for uniquely defined (Xf (A),Yf ( )) ∈ Ω (Σ, ad(P )) × Ω (Σ, ad(P )). (For later convenience, we put a minus sign in front of Xf (A)). We define the perturbed elliptic Yang–Mills flow to be the system of equations

0= ∂ A − d Ψ − ∗d ω − X (A), (32) s A A f

(0= ∂sω + [Ψ,ω] − ω + ∗FA + Yf (A). Equations (32) are invariant under the action of time-dependent gauge transformations in G(P ). Since in the following all statements are formulated in a gauge-invariant way we may always assume that Ψ = 0. 6.2. Perturbed critical point equation. The above defined class of holonomy perturbations will turn out to be large enough to achieve surjectivity of the linearized operators D(A,ω,Ψ) along connecting trajectories (A, ω, Ψ). However, in order to achieve nondegeneracy of critical points of the perturbed functional J + hf it is sufficient to work with the more restricted class of holonomy perturbations which depend on A (but not on ω). Such perturbations are maps hf : A(P ) → R defined as in (30) where we set

A := A + 0 dt. In this case the (negative of the) gradient of hf at A is the 1 uniquely deﬁned Xf (A) ∈ Ω (Σ, ad(P )) such that

(33) dhf (A)(α)= h−Xf (A) ∧ ∗αi ZΣ for all α ∈ Ω1(Σ, ad(P )). Critical points of the thus perturbed elliptic Yang– Mills functional J + hf are solutions (A, ω) of the system of equations 0= ∗d ω + X (A), (34) A f (0= ∗FA − ω.

The set of critical points of the functional J +hf is denoted by Crit(J +hf ). For a> 0 we also set a Crit (J + hf ) := {(A, ω) ∈ Crit(J + hf ) |J (A, ω) < a}. 22 R.JANNERANDJ.SWOBODA

J ,f 1,p p We deﬁne the linear operator H(A,ω) : W (Σ) → L (Σ) to be the Hessian at (A, ω) of the perturbed functional J + hf , i.e. J ,f H(A,ω)(α, v) = (∗dAv − ∗[ω ∧ α]+dXf (A)α, ∗dAα − v).

Note that the previously deﬁned operator B(A,ω,0) (cf. (6)) is the Hessian J ,0 H(A,ω) considered here, augmented by a gauge-ﬁxing condition. Associated with each (A, ω) ∈ Crit(J + hf ) is the twisted de Rham complex

(d ,[ω∧· ]) Ω0(Σ, ad(P )) A−→ Ω1(Σ, ad(P )) ⊕ Ω0(Σ, ad(P ))

J ,f H(A,ω) −→ Ω1(Σ, ad(P )) ⊕ Ω0(Σ, ad(P )). The first operator in this complex is the infinitesimal action of the group G(P ) at (A, ω). Let ker HJ ,f 0 1 (A,ω) H(A,ω) := ker(dA, [ω ∧ · ]),H(A,ω,f) := im(dA, [ω ∧ · ]) be the cohomology groups arising from that complex. We call a critical 0 point (A, ω) ∈ Crit(J + hf ) irreducible if H(A,ω) = 0. It is called nonde- 1 generate if H(A,ω,f) = 0. 6.3. Transversality theorem. Throughout this section we fix a number a> 0.

Definition 15. Let m ∈ N. A pair (γ,f) ∈ Γm × Fm is called a-regular if the following two conditions are satisfied. a (i) Every critical point A = (A, ω) ∈ Crit (J + hf ) is irreducible and nondegenerate. 0 (ii) Let A = (A, ω): R →A(P )×Ω (Σ, ad(P )) be a solution of Eq. (32) such ± ± ± ± a that lims→±∞(A(s),ω(s)) = (A ,ω ) for a pair (A ,ω ) ∈ Crit (J +hf ). Then the operator D(A,ω,0) defined in (6) is surjective (for every p> 1). a For γ ∈ Γm we denote by Freg(γ) the set of maps f ∈ Fm such that (γ,f) is a-regular. We now proceed in two steps. We first show that nondegeneracy of critical points in the sense of Definition 15 (i) can be achieved by adding to J a perturbation hf which depends only on A. Thereafter we show that there exists an a-regular perturbation hf ′ (which now may depend on (A, ω)) close a a to hf such that Crit (J + hf ) = Crit (J + hf ′ ). To be able to formulate our transversality theorems it is necessary to first introduce a family of seminorms on the space of perturbations. Here we follow again closely [21]

and deﬁne for any k ≥ 1 and perturbation f the k-seminorm of Xf as A kXf (A)kCk k dXf ( )(α + v dt)kCk−1

kXf kk := sup + sup k−1 . A ˜ 1+ kAk k ˜ kα + v dtk k−1 (1 + k k k−1 ) A C C C ∈A(P ) α+v dt∈TA A(P ) ! ELLIPTIC YANG–MILLS FLOW THEORY 23

A k-seminorm for Yf is deﬁned analogously. In the case where Xf depends only on A (and not on ω), we modify the deﬁnition of kXf kk accordingly.

Theorem 16. Let (γ0,f0) ∈ Γm0 × Fm0 be such that every critical point of

J + hf0 in the sublevel set of a is irreducible. Then, for every k ≥ 1 and ε> 0, there exists n ∈ N and a pair (γ,f) ∈ Γn × Fn that satisﬁes condition

(i) in Deﬁnition 15 and such that kXf − Xf0 kk < ε. Proof. We split the proof into ﬁve steps.

Step 1. Assume the connection A ∈A(P ) is irreducible. Let hf : A(P ) → R be a perturbation such that the kernel of the perturbed Yang–Mills Hessian ∗ 0 HA,f : α 7→ dAdAα + ∗[∗FA ∧ α]+dXf (A)α equals im dA :Ω (Σ, ad(P )) → Ω1(Σ, ad(P )) . Then h satisfies condition (i) in Definition 15. f It suffices to show that the assumptions imply that the operator Bf,0 as defined in Proposition 5 is injective. By Proposition 7 applied with λ = 0, injectivity of Bf,0 is equivalent to injectivity of the operator Cf,0 = Cf . a Since (A, ω) ∈ Crit (J + hf ) and therefore ω = ∗FA, the operator Cf takes ± the form of the operators Cf considered in the proof of Lemma 9. But for these injectivity has been shown under the same assumptions we made here, and hence the claim follows.

Step 2. Let (γ0,f0) ∈ Γm0 × Fm0 satisfy the assumptions of the theorem. Then there exists m>m0 and γ ∈ Γm with γi = γ0i for i = 1,...,m0 such that the following condition is satisfied. Let m σ(A) := ρ(0, A) = (ρ1(0, A), . . . , ρm(0, A)) ∈ G . a Then for every critical point (A, ω) ∈ Crit (J + hf0 ) and every nonzero ∗ m α ∈ ker HA,f0 such that dAα = 0, it follows that [dσ(A)α] ∈ T[σ(A)](G /G) is nonzero. ∗ For any fixed nonzero α ∈ ker HA,f0 such that dAα = 0 it follows from statement (ii) in Proposition 18 that we can extend γ0 to γ such that [dσ(A)α] 6= 0. Because the latter property is open it remains satisfied for all α′ sufficiently close to α. Elliptic theory implies that the unit sphere in ker H | ∗ is compact. A standard compactness argument now com- A,f0 ker dA pletes the proof of the claim.

Step 3. Let m ∈ N and (γ,f) ∈ Γm ×Fm. Fix a number p> 1 and let q > 1 with p−1 + q−1 = 1 denote the Sobolev exponent dual to p. Then surjectivity 1,p p of the operator HA,f : W (Σ) → L (Σ) is equivalent to the vanishing of q ∗ every β ∈ L (Σ) with the property that β ∈ ker HA,f and

(35) hXf (A) ∧ ∗βi = 0. ZΣ As follows from standard arguments, the range of HA,f is closed. Then surjectivity of HA,f is equivalent to injectivity of its dual operator, whence the claim. 24 R.JANNERANDJ.SWOBODA

Step 4. Let (γ0,f0) and m>m0 be as in Step 2. For k ∈ N we denote k k+1 D m G Fm := C ( × G ) and consider the linear operator 1,p k p ′ (36) W (Σ) × Fm → L (Σ): (α, f ) 7→ HA,f α + Xf ′ (A).

Then for every k ∈ N there exists ε > 0 such that for all f ∈ Fm with a kXf0 − Xf kk < ε and for all (A, ω) ∈ Crit (J + hf ) the operator in (36) is surjective.

We first verify the claim for f = f0. Assume by contradiction that there a exists (A, ω) ∈ Crit (J +hf0 ) such that the operator in (36) is not surjective. Then by Step 3 there exists β 6=0 in Lq(Σ) which satisfies β ∈ ker H∗ as A,f0 m well as (35). It follows from Step 2 that [dσ(A)β] 6=0 in T[σ(A)]G /G. This implies that the map r 7→ [ρ(0, A + rβ)] is an embedding of a sufficiently ˆ k small neighbourhood of zero. Therefore there exists a map f ∈ Fm such that fˆ(z, ρ(z, A + rβ)) = rχ(r)χ(|z|) is satisfied for all r ∈ R and all z ∈ D. Here χ: R → [0, 1] is a suitably chosen smooth cutoff function such that χ equals 1 near 0 and supp χ is contained in a sufficiently small neighbourhood of 0. It follows that d ˆ 2 2 dhfˆ(A)β = f(z, ρ(z, A + rβ)) d z = χ(|z|) d z > 0. dr r=0 D D Z Z This inequality contradicts Eq. (35). The claim in the case f = f0 follows. a Because surjectivity is an open condition and the set Crit (J + hf0 )/G(P ) is compact it follows that the operator in (36) is surjective for all f ∈ Fm such that kXf − Xf0 kk is sufficiently small. Step 5. We prove the theorem. We first claim that for every k ∈ N there exists ε> 0 such that

∗ k,ε M (Fm ) := ∗ {(A, f) ∈A(P )×Fm | dAFA+Xf (A) = 0, J (A, ∗FA) < a, kXf −Xf0 kk < ε} k ∗ is a C -Banach manifold. Namely, the linearization of the equation dAFA + ∗ k,ε Xf (A) = 0 at each (A, f) ∈ M (Fm ) is a surjective operator as was shown in Step 4. It is Fredholm by standard arguments. Therefore the inﬁnite- dimensional version of the implicit function theorem proves the claim. It ∗ k,ε k,ε k follows that the projection map π : M (Fm ) → Fm is a C -Fredholm map between Ck-Banach manifolds. The Fredholm index of its linearization ∗ k,ε dπ(A, f) is zero for all (A, f) ∈ M (Fm ). Hence the Sard–Smale theorem k,ε applies and shows that the set of regular values of π is dense in Fm . For each such regular value f and every A ∈ π−1(f) it follows from surjectivity of the operator in (36) and since each dπ(A, f) has Fredholm index equal to zero 1 that the perturbed Hessian HA,f is surjective. This shows that H(A,ω,f) = 0 a for all (A, ω) ∈ Crit (J + hf ). Because irreducibility of connections is an ELLIPTIC YANG–MILLS FLOW THEORY 25

0 open condition we also have that H(A,ω) = 0 for all such (A, ω) after choosing ε > 0 still smaller if necessary. That in fact f can be chosen to be smooth follows from a density argument as in [21, Theorem 8.3]. This completes the proof of Theorem 16.

Theorem 17. Let (γ0,f0) ∈ Γm0 ×Fm0 be the perturbation as in the conclu- sion of Theorem 16 (which was named (γ,f) there). Then for every k ≥ 1 and ε> 0 there exists m ≥ 1 and an a-regular pair (γ1,f1) ∈ Γm ×Fm which satisﬁes a a (37) Crit (J + hf0 ) = Crit (J + hf0 + hf1 ), a (38) (A, ω) ∈ Crit (J + hf0 ) =⇒ hf1 (A, ω) = 0,

(39) kXf1 kk < ε. Proof. We proceed in three steps. At several places we identify (A, ω) with

A = A + ω dt as explained above.

Step 1. There exists m ≥ m0 and γ ∈ Γm with γi = γ0i for i = 1,...,m0 such that the following condition is satisﬁed. Let

A A σ(A) := ρ(0, A) = (ρ1(0, ), . . . , ρm(0, )) ∈ G . ± ± a For every pair of distinct critical points (A ,ω ) ∈ Crit (J +hf0 ) and every − − + + R (A, ω, 0) ∈ Mf0 (A ,ω , A ,ω ) (cf. §3 for notation) there is s0 ∈ such that the following holds. First,

c a

A A (40) σ(A(s0)) ∈/ {σ( ) | = A + ω dt, (A, ω) ∈ Crit (J + hf0 )}.

∗ A Second, for every nonzero η ∈ ker D(A,ω,0) it follows that [dσ(A(s0))∂s (s0)]

m A and [dσ( (s0))η(s0)] are linearly independent vectors in T[σ(A(s0))]G /G.

Fix s0 ∈ R such that the connection A(s0) is irreducible. Such a choice ± ± of s0 is possible because the limit connections (A ,ω ) are irreducible by assumption. Since (A+,ω+) and (A−,ω−) are distinct it follows that − − + + (J + hf0 )(A ,ω ) < (J + hf0 )(A(s0)) < (J + hf0 )(A ,ω ). Therefore + + − − A(s0) is not gauge equivalent to (A ,ω ) or to (A ,ω ). It then follows from Proposition 18 (i) that there exist m ≥ m0 and γ ∈ Γm as required and such that (40) holds. Furthermore, it follows from statement (ii) in

Proposition 18 that we can choose m ∈ N and γ ∈ Γm such that in addition A

dσ(A(s0))∂s (s0) ∈/ V . Here we let A V := {(ξσi(A(s0)) − σi( (s0))ξ)i=1,...,m | ξ ∈ g}

denote the tangent space at (σi(A(s0))i=1,...,m to its G-orbit by simultaneous A conjugation. It follows that δ := infv∈V kdσ(A(s0))∂s (s0) − vk > 0. By ∗ compactness of the unit sphere in ker D(A,ω,0) it suffices to prove the claim ∗ for each fixed nonzero η ∈ ker D(A,ω,0). Note that η(s0) 6= 0 by a unique continuation argument. By definition of the constant δ it follows that

(41) dσ(A(s0))(η(s0)+ λ∂sA(s0)) ∈/ V 26 R.JANNERANDJ.SWOBODA

−1 for all |λ| > c := δ kdσ(A(s0))η(s0)k. Hence for this range of λ we see that

m A [dσ( (s0))(η(s0)+ λ∂sA(s0))] 6= 0 in T[σ(A(s0))]G /G. To also show this property for |λ| ≤ c it suffices to prove it for any fixed such λ by openess of the condition (41) and compactness of the interval [−c, c]. We fix an arbitrary λ ∈ [−c, c]. Because η(s0) 6= 0 and ∂s A(s0) 6= 0 it follows from

Proposition 19 that η(s0) and ∂s A(s0) are linearly independent, and hence η(s0)+ λ∂sA(s0) 6= 0. Again by Proposition 19 it follows that d∗ (η(s )+ λ∂ A(s )) = 0. A 0 s 0

Thus Proposition 18 yields the existence of a number m1 ≥ m and a suit-

able tuple (γ1, . . . , γm1 ) ∈ Γm1 of embedded loops (with γi as before for A i = 1,...,m) such that [dσ(A(s0))(η(s0) − λ∂s (s0))] does not vanish, as desired. This completes the proof of the claim.

Step 2. For k ∈ N and ε1 > 0 we denote

k,ε1 k+1 m G F := {f ∈ C (D × G ) | f| = 0, kfk k+1 < ε }, m Bε1 (C) C 1

m a A

A D where C := {(z, ρ(z, )) ∈ × G | = (A, ω) ∈ Crit (J + hf0 )}. Then for every k ∈ N the constant ε1 > 0 can be chosen such that conditions (37), k,ε1 (38), (39) are satisﬁed for every perturbation f ∈ Fm . Furthermore, for ± ± a k,ε1 every (A ,ω ) ∈ Crit (J + hf0 ) and f ∈ Fm the linear operator ˆ p p D(A,ω) : W ⊕ Tf Fm →L ,

′ A (α,v,ψ,f ) 7→ D(A,ω,0)(α, v, ψ) + (−Xf ′ (A),Yf ′ ( ), 0) − − + + is surjective for all (A, ω, 0) ∈ Mf (A ,ω , A ,ω ). It follows by definition of the space F k,ε1 that conditions (37), (38), (39) c m are satisfied for every sufficiently small ε1 < ε. To prove the second claim ˆ we first note that by standard arguments the range of D(A,ω,f) is closed. Let q > 1 with p−1 + q−1 = 1 denote the Sobolev exponent dual to p. ˆ Then surjectivity of D(A,ω,f) is equivalent to the vanishing of every η ∈ q ˆ L orthogonal to im D(A,ω,f). Any such η = (η0, η1, η2) is contained in ˆ∗ ker D(A,ω,f) and satisfies

∞ A (42) hXf ′ (A(s)), η0(s)i + hYf ′ ( (s)), η1(s)i ds = 0 Z−∞ ′ for every f ∈ Tf Fm. We first prove the assertion for f = f0 and a fixed pair ± ± a (A ,ω ) ∈ Crit (J + hf0 ). Assume by contradiction that η = (η0, η1, η2) 6= 0 satisfies both Dˆ∗ η = 0 and (42). As follows from Step 1 there

(A,ω,f0)

A A exists s0 ∈ R such that [dσ(A(s0))∂s (s0)] and [dσ( (s0))η(s0)] are linearly m independent vectors in T[σ(A(s0))](G /G). Therefore the map m

(r, s) 7→ [ρ(z, A(s)+ rη(s))] ∈ G /G 2 is an embedding of a neighbourhood of (0,s0) ∈ R . Now let χ: R → [0, 1] be a suitably chosen smooth cutoff function such that supp χ is contained in ELLIPTIC YANG–MILLS FLOW THEORY 27 a sufficiently small neighbourhood of 0 and is equal to 1 near 0. Then there exists a smooth G-invariant function f ′ : D × Gm → R which by Step 1 can be chosen to vanish in a neighbourhood of C and satisfies ′ f (z, ρ(z, A(s)+ rη(s))) = rχ(r)χ(s − s0)χ(|z|). for all (z,r,s) ∈ D × R × R. For each s ∈ R this implies that

d ′ 2 A dhf ′ (A(s))η(s) = f (z, ρ(z, (s)+ rη(s))) d z dr r=0 D Z 2 = χ(s − s0) χ(|z|) d z > 0. D Z ′ Hence there exists a perturbation f ∈ Tf0 Fm for which (42) is not satisﬁed. ˆ This contradiction shows that η = 0. Therefore the operator D(A,ω,f0) is ± ± a surjective. Surjectivity for every pair (A ,ω ) ∈ Crit (J + hf0 ) and f ∈ k,ε1 Fm follows from a compactness argument after choosing ε1 suﬃciently small. Step 3. We prove the theorem. The statement of the theorem follows from Step 2 by carrying over the arguments of Step 5 of the proof of Theorem 16 to the present situation. In the course of proof of the transversality theorems we made frequently use of the following two auxiliary results. 1 1 Proposition 18. For an embedding γ : S → Σ × S let ργ : A˜(P ) → G ˜ denote the resulting holonomy map, asigning A ∈ A(P ) its holonomy along γ. Then the following statements hold true.

˜ A (i) Two connections A1 = A1 + ω1 dt, 2 = A2 + ω2 dt ∈ A(P ) are gauge equivalent if and only if there exists g ∈ G such that

−1 A ργ (A2)= g ργ( 1)g for every embedding γ : S1 → Σ × S1. ˜ 1 0 (ii) Let A = A + ω dt ∈ A(P ) and (α, v) ∈ Ω (Σ, ad(P )) ⊕ Ω (Σ, ad(P )). Then 0 1 1 1

α + v dt ∈ im dA :Ω (Σ, ad(P )) → Ω (Σ × S , ad(P × S )

(where we denotedA = dA +[ω∧ · ]∧ dt) if and only if for every embedding

γ : S1 → Σ × S1 there exists ξ ∈ g such that

A A dργ(A)(α + v dt)= ργ( )ξ − ξργ ( ). ˜ Proof. Both statements follow from the fact that a connection A ∈ A(P ) is uniquely determined by parallel transport along the set of embedded loops in Σ × S1.

Proposition 19. Assume (A, ω, 0) is a solution of Eq. (32). Denote as A before = A + ω dt and dA = dA + [ω ∧ · ] ∧ dt. Then (η0(s)+ η1(s) dt) ⊥ ∗ R im dA(s) for all η = (η0, η1, 0) ∈ ker D(A,ω,0) and s ∈ . The same relation

28 R.JANNERANDJ.SWOBODA A holds true for η = ∂s A. Furthermore, hη0(s)+ η1(s) dt, ∂s (s)i = 0 for all ∗ R η = (η0, η1, 0) ∈ ker D(A,ω,0) and s ∈ . ∗

Proof. Let (η0, η1, 0) ∈ ker D(A,ω,0). Denotingη ˆ := (η0, η1) and A B(A,ω)(α, v) := (∗[ω ∧ α] − ∗dAv − dXf (A)(α, v), ∗dAα − v +dYf ( )(α, v))

this implies that −∂sηˆ + B(A,ω)ηˆ = 0. On the other hand, from gauge- A invariance of Eq. (32) it follows that ∂sdA ϕ + B(A,ω)d ϕ = 0. Therefore, by symmetry of the operator B(A,ω) it follows that

A A hη,ˆ dA ϕi =h∂ η,ˆ d ϕi + hη,ˆ ∂ d ϕi

ds s s A =hB(A,ω)η,ˆ dA ϕi + hη,ˆ −B(A,ω)d ϕi =0.

This shows that the inner product hη,ˆ dA ϕi is constant in s. Becauseη ˆ(s) converges to zero as s → ±∞ this constant must be zero and the ﬁrst claim follows. Let ϕ ∈ Ω0(Σ, ad(P )). Then the second claim follows from the calculation

h∂sA + ∂sω dt,dAϕ + [ω ∧ ϕ] ∧ dti = h∂sA, dAϕi + h∂sω, [ω ∧ ϕ]i A =h∗dAω + Xf (A), dAϕi + hω − ∗FA − Yf ( ), [ω ∧ ϕ]i

∗ ∗ A =hdA ∗ dAω + dAXf (A), ϕi + h∗[ω ∧ FA] + [ω ∧ Yf ( )], ϕi =0. The second equation holds by (32) and the fourth equation follows from

∗ A dAXf (A) = 0, [ω ∧ Yf ( )] = 0, ∗ together with the identity dA ∗ dAω = ∗dAdAω = ∗[FA ∧ ω]. This proves the d second claim. Finally, because ∂s A satisfies the linearized equation ( ds + d B(A,ω))∂s A = 0 andη ˆ = (η0, η1) satisfies by assumption (− ds +B(A,ω))ˆη = 0 it follows by the same calculation as in the proof of the first claim that hηˆ(s), ∂s A(s)i = 0 for all s ∈ R.

7. Elliptic Yang–Mills homology We fix a closed oriented Riemannian surface (Σ, g), a compact Lie group G, a principal G-bundle P → Σ, a regular value a> 0 of J , and an a-regular perturbation hf . Associated to these data we define the elliptic Yang–Mills homology HY M∗(Σ, P, g, a, f) as follows. Let Crita(J + h ) Ra := f G(P ) denote the (finite) set of gauge equivalence classes of critical points of J +hf of energy less than a. These generate a chain complex a CY M∗ (Σ,P,g,f) := Z2h[(A, ω)]i a [(A,ωX)]∈R ELLIPTIC YANG–MILLS FLOW THEORY 29 with grading given by the index ind HA,f of the perturbed Yang–Mills Hes- sian (cf. Theorem 4 for further details). For a pair [(A−,ω−], [(A+,ω+)] ∈ Ra of critical points we let M (A−,ω−, A+,ω+) M (A−,ω−, A+,ω+)= f f G(P ) be the moduli space as defined in §3.c It is a smooth manifold of dimension R ind HA−,f − ind HA+,f . The group acts on it freely by time-shifts. If this index difference equals 1 it follows by compactness (cf. Theorem 11) that − − + + the quotient of Mf (A ,ω , A ,ω ) modulo the R-action consists of a finite number of points. For each k ∈ Æ0 we hence obtain a well-defined boundary operator a a ∂k : CY Mk (Σ,P,g,f) → CY Mk−1(Σ,P,g,f) as the linear extension of the map ′ ′ ∂kx := n(x,x )x , x′∈Ra ind(Xx′)=k−1 a ′ x ∈ R a critical point of index ind x = k. Here n(x,x ) ∈ Z2 is defined for x = [(A−,ω−)] and x′ = [(A+,ω+)] as the number of R-equivalence classes − − + + of elements in Mf (A ,ω , A ,ω ), counted modulo 2, i.e. − − + + ′ Mf (A ,ω , A ,ω ) n(x,x ) :=# R (mod 2).

Lemma 20. The sequence ∂∗ of homomorphisms satisfies ∂∗ ◦ ∂∗+1 = 0, a i.e. (CY M∗ (Σ,P,g,f), ∂∗) is a chain complex. Proof. The property of a chain map follows from standard arguments invok- ing Theorems 11 (compactness) and 14 (exponential decay of finite energy gradient flow lines). We define the elliptic Yang–Mills homology at level a associated with the data (Σ,P,g,f) to be the collection of abelian groups

a ker ∂k HY Mk (Σ,P,g,f) := (k ∈ Æ0). im ∂k+1 We conclude this section with a number of remarks. • We expect the usual cobordism arguments to show independence of the a homology groups HY M∗ (Σ,P,g,f) on the choice of the Riemannian metric g and a-regular perturbation f used to define it. • To keep the exposition short we developed here a version of elliptic Yang– Mills homology using coefficients in Z2. Defining a variant of it with coefficient ring Z requires to deal with oriented moduli spaces. • It would be very interesting to compare elliptic Yang–Mills homology with the Morse homology obtained from the Yang–Mills gradient flow on a Rie- mannian surface and first considered by Atiyah and Bott [3]. The chain 30 R.JANNERANDJ.SWOBODA

complexes in both cases are generated by (perturbed) Yang–Mills connections A (respectively pairs (A, ω) where A is a (perturbed) Yang–Mills connection and ω = ∗FA), and hence coincide. However the boundary operators are very diﬀerent, involving a parabolic equation in the classical situation, in contrast to the elliptic system in the approach presented here. We shall comment more on this point in §8.

8. Relationship to parabolic Yang–Mills Morse homology We now discuss a partly conjectural relationship between elliptic Yang– Mills homology and the by now classical Morse homology theory based on the L2-gradient ﬂow of the Yang–Mills functional over Riemann surfaces. The latter will be called parabolic Yang–Mills homology for brevity. We refer to the seminal article [3] by Atiyah and Bott which initiated (amongst other things) the study of the Yang–Mills functional on Riemann surfaces from a Morse theoretical perspective. Recall that the (unperturbed) Yang– Mills functional is the map 1 YM: A(P ) → R, YM(A)= |F |2 dvol . 2 A Σ ZΣ The following observation provides the starting point of our discussion. Lemma 21. For all (A, ω) ∈ A(P ) × Ω0(Σ, ad(P )) there holds the energy inequality (43) J (A, ω) ≤ YM(A), with equality if and only if ω = ∗FA. Proof. The claim follows, using the Cauchy–Schwartz inequality, from 1 1 J (A, ω)= hF , ∗ωi− |ω|2 dvol ≤ |F ||ω|− |ω|2 dvol A 2 Σ A 2 Σ ZΣ ZΣ 1 ≤ |F |2 dvol = YM(A). 2 A Σ ZΣ The case of equality can easily be read oﬀ from this estimate. We now consider the G(P )-equivariant map 0 θ : A(P ) →A(P ) × Ω (Σ, ad(P )), A 7→ (A, ∗FA). As initially remarked, a pair (A, ω) is a critical point of J if and only if A is a Yang–Mills connection and ω = ∗FA. Hence the map θ induces a G(P )- equivariant bijection between the sets Crit(YM) and Crit(J ). By Lemma 21 it decreases energy in the sense that (44) J ◦ θ(A) ≤ YM(A) for all A ∈ A(P ). This observation plays a crucial role in the following construction. The idea is to couple solutions of the Yang–Mills gradient ELLIPTIC YANG–MILLS FLOW THEORY 31

flow equation ∗ (45) 0= ∂sA + dAFA on the half-infinite interval (−∞, 0] with solutions of the (unperturbed) equation (3) on the half-infinite interval [0, ∞). We are thus lead to define for critical points A− ∈ Crit(YM) and (B+,ω+) ∈ Crit(J ) the set

− + + Mhyb(A ; B ,ω ) := (A; B,ω) | A satisfies (45) on (−∞, 0) × Σ, lim A(s)= A−, c s→−∞ (nB,ω) satisfies (3) on (0, ∞) × Σ, lim (B(s),ω(s)) = (B+,ω+), s→∞ (B(0),ω(0)) = θ(A(0)) , and call the quotient o M (A−; B+,ω+) M (A−; B+,ω+) := hyb hybr G(P ) c hybrid moduli space (of connecting trajectories between A− and (B+,ω+)). We refer to the articles [1, 24] for related constructions of hybrid moduli spaces in various different contexts. The significance of the moduli spaces − + + Mhybr(A ; B ,ω ) introduced here is that they are supposed to give rise to a chain map Θ between the Morse complexes defined from the elliptic and the parabolic Yang–Mills flow equations. Namely, after introducing a suitable perturbation scheme (which for the Yang–Mills gradient flow has been done in [23]) we expect these hybrid moduli spaces to be smooth manifolds of dimension equal to the index difference ind HA− − ind HB+ of the corresponding Yang–Mills Hessians. As in [1, 24] they are expected to admit compactifications by adding configurations of broken trajectories. In partic- − + + ular, in the case of index difference equal to zero, Mhybr(A ; B ,ω ) is a finite set and we may then define # Mhybr(A−; B+,ω+) to be the number of elements of Mhybr(A−; B+,ω+) (again counted modulo 2 to avoid orientation issues). For the remainder of this section we fix a regular value a> 0 of YM and let Crita(YM) denote the set of critical points A of YM with YM(A) ≤ a. By the above considerations, we obtain a well-defined map

Θ := (Θk)k∈Æ0 , where the homomorphism Θk is deﬁned by linear extension of the map

− − + + + + Θk(A ) := # Mhybr(A ; B ,ω ) · (B ,ω ), (B+,ω+)∈Crita(J ) ind H =k XB+ − a where A ∈ Crit (YM) with ind HA− = k. By standard arguments, the map Θ is a chain map between the Morse complexes generated by the sets Crita(YM) and Crita(J ). The crucial observation, ﬁrst made in a related situation in [1], cf. also [24], is that the homomorphism Θ in each degree k 32 R.JANNERANDJ.SWOBODA can be represented by an invertible upper-triangular matrix of the form 1 ∗ ··· ··· ∗ .. .  0 1 . .  (Θk ) = ...... ∈ Zm×m, ij 1≤i,j≤m  . . . . .  2  . .   . ..   . 1 ∗     0 ··· ··· 0 1    Zm×m and hence is invertible in 2 . The existence of such an invertible upper- k triangular matrix (Θij)1≤i,j≤m can be seen as follows. As a set of generators of the chain group of degree k ∈ N0 of the parabolic Yang–Mills Morse a complex we choose the critical points (A1,...,Am) ∈ Crit (YM) of index ind HAj = k, ordered by increasing action. We then deﬁne the m-tuple

((B1,ω1),..., (Am,ωm)) := (θ(A1),...,θ(Am)), which as explained above is a set of generators of the chain group of degree k of the elliptic Yang–Mills Morse complex. Inequality (44) (which in this case is an equality) moreover shows that this m-tuple is ordered by increasing action, too. It follows from Lemma 21 and inequality (45) by an elementary argument that − + + (46) Mhybr(Ai ; Bj ,ωj )= ∅ if i < j. − + + Namely, along any ﬂow line in Mhybr(Ai ; Bj ,ωj ) energy is strictly increasing and therefore inequality (45) implies (46). Furthermore, each moduli − + + space Mhybr(Ai ; Bi ,ωi ) is representedc by precisely one element, the con- − + + catenation of the constant ﬂow lines A ≡ Ai and (B,ω) ≡ (Bi ,ωi ). The k invertible chain homomorphism Θ descends in each degree k ∈ N0 to an isomorphism between the k-th elliptic and parabolic Yang–Mills homology groups (at level a). The details of the construction of this isomorphism are similar to those being carried out in [1, 24]. We leave them to a forthcoming publication.

9. Three-dimensional product manifolds So far, we deﬁned the elliptic Yang–Mills homology in the case n = 2 which allows the equations to be nicely elliptic. In the subsequent sections we discuss two extensions of this theory to the case n ≥ 3. First we explain how two-dimensional elliptic Yang–Mills homology may be applied to the special case of three-dimensional products Y =Σ×S1. Then in §10 we show how to restore ellipticity by restricting the space X = A(P )×Ωn−2(Y, ad(P )) to a suitable Banach submanifold.

We start by recalling some facts concerning the moduli space Mγ (P ) of ﬂat connections on a principal bundle P over a Riemann surface (Σ, gΣ) of genus γ. It was investigated for the ﬁrst time in 1983 by Atiyah and ELLIPTIC YANG–MILLS FLOW THEORY 33

Bott in [3]. They noticed that the conformal structure of Σ gives rise to a Riemannian metric and an almost complex structure on Mγ (P ); moreover, if a nontrivial principal SO(3)-bundle P is chosen, then the moduli space Mγ (P ), defined as the quotient of the space of the flat connections A0(P ) ⊆ A(P ) and the identity component G0(P ) of the group of gauge transformations, is a compact Kähler manifold without singularities of dimension 6γ − 6 (cf. [7]). In the nineties some aspects of the topology of Mγ(P ) were investigated by Dostoglou and Salamon (cf. [7, 8, 9, 19]), who proved among other results the Atiyah–Floer conjecture in this context, as well as by Hong (cf. [11]).

1 We fix a three-dimensional product manifold Y =Σ×S , (Σ, gΣ) a Riemann 2 surface of genus γ, with the partially rescaled product metric ε gΣ ⊕ gS1 , for a parameter ε > 0. We include in the discussion here only nontrivial principal SO(3)-bundles P and Pˆ = P ×S1 over Σ and Y , respectively. The motivation for considering this setup comes from the fact that, as we will see, the perturbed Yang–Mills connections on P and the resulting elliptic Yang–Mills homology groups are strongly related to the perturbed geodesics and the homology of the free loop space LMγ(P ) associated with the moduli space of flat connections Mγ (P ). In fact, by results of the first author (cf. [14]), there is a bijection between the perturbed Yang–Mills connections on the bundle Pˆ and the perturbed geodesics on Mγ(P ). Furthermore, he showed that the Morse homologies, defined from the perturbed parabolic Yang–Mills gradient flow on Pˆ → Σ × S1 and the heat flow on Mγ (P ), are isomorphic, provided that ε is small enough and that an energy bound b is chosen (cf. [12, 13]). Hence we have:

b γ ε,b (47) HM∗ L M (P ), Z2 =∼ HM∗ A (Pˆ)/G0(Pˆ), Z2 where LbMγ (P ) ⊆ LMγ(P ) and Aε,b(Pˆ) ⊆A(Pˆ), respectively, denote the subsets with energies bounded from above by b.

The heat flow homology appearing on the left-hand side of (47) is well- defined in any dimension due to work of Salamon and Weber (cf. [20, 29, 30]). In contrast, the Yang–Mills gradient flow exists only if the base manifold Y is two- or three-dimensional, or if it has a symmetry of codimension three (cf. [15, 16]). Furthermore, apart from the two-dimensional case, the Morse–Smale transversality is not yet proven and thus the homology ε,b HM∗ A (P )/G0(P ), Z2 is presently not defined in the general case. One notable exception are products Y = Σ × S1. Here the gauge equivalence classes of gradient flow lines are in a bijective relation with the gauge equivalence classes of heat gradient flow lines on LMγ(P ) provided that ε is small ε,b enough and thus, the homology HM∗ A (P )/G0(P ), Z2 is well defined.

As follows from works of Viterbo (cf. [26]), Salamon and Webe r (cf. [20]) and 34 R.JANNERANDJ.SWOBODA

Abbondandolo and Schwarz (cf. [1, 2]), the Morse homology of LbMγ (P ) is isomorphic to Floer homology of the cotangent bundle T ∗Mγ (P ) defined via the Hamiltonian HV given by the sum of kinetic and potential energy and considering only orbits with action bounded by b. Moreover, Weber (cf. [29]) showed that the Morse homology of the free loop space defined from the heat flow is isomorphic to its singular homology. Summarizing we therefore have (in part conjecturally) the following isomorphisms of abelian groups: b γ ε,b H∗ L M (P ) H∗ A (Pˆ)/G0(Pˆ) ? =∼ =∼ b γ ε,b HM∗ L M (P ) =∼ HM∗ A (Pˆ)/G0(Pˆ) =∼ b ∗ γ HF∗ T M (P ),HV It is still an open question (marked with an interrogation point in the above ε,b diagram) whether the Morse homology of A (Pˆ)/G0(Pˆ) defined from the L2-gradient flow of the Yang–Mills functional is isomorphic to the singular homology. We may also ask how our newly defined elliptic Yang– Mills homology fits into this picture. To be specific, we consider the el- b 1 2 liptic Yang–Mills homology HY M (Σ × S , ε gΣ ⊕ gS1 ,f) defined from the 1 1 functional J + hf on A(Pˆ) × Ω (Σ × S , ad(P )), which can be seen as the Yang–Mills analogue of the cotangent bundle. In this special case it might be possible to show a bijective relation between the gauge equivalence classes of the elliptic Yang-Mills flow lines between stationary points and those of the parabolic Yang-Mills or of the Floer homology setup, provided that ε is small enough, by an analogous argument as in [12] or in [20]. We leave the proof to a forthcoming publication. This would then de- b 1 2 fine HY M (Σ × S , ε gΣ ⊕ gS1 ,f) and lead to an isomorphismus between it, ε,b ˆ ˆ b ∗ γ Z HM∗ A (P )/G0(P ) and HF∗ T M (P ),HV , considering 2 coefficients homologies: b γ H∗ L M (P ) =∼ b γ ε,b HM∗ L M (P ) =∼ HM∗ A (Pˆ)/G0(Pˆ) ? =∼ =∼ ? b ∗ γ ∼ b 1 2 HF∗ T M (P ),HV = HY M (Σ × S , ε gΣ ⊕ gS1 ,f) 10. Restricted flow In this final section we discuss a modification and extension of Eq. (3) to base manifolds Y of dimension n ≥ 3. It is motivated from the shortcoming that the linearization of Eq. (3) is an elliptic system only in dimension n = 2, even after imposing gauge-fixing conditions. ELLIPTIC YANG–MILLS FLOW THEORY 35

Proposition 22. Let I be an open interval and (A, ω, Ψ) ∈ C∞(I, A(P ) × n−2 0 Ω (Y, ad(P )) × Ω (Y, ad(P ))). Then the linear operator ∇s + B(A,ω,Ψ) associated with (A, ω, Ψ) (cf. (6)) is elliptic if and only if n = 2. Proof. It is straightforward to verify the assertion from a calculation of the principal symbol of the operator ∇s +B(A,ω,Ψ). As this requires to introduce some notation we instead prove the claim for the operator ∗ 2 2 (∇s + B(A,ω,Ψ))(∇s + B(A,ω,Ψ)) = −∇s + B(A,ω,Ψ). A short calculation shows that 2 −∇s +∆A 0 0 2 2 2 ∗ −∇s + B(A,ω,Ψ) = 0 −∇s + dAdA 0 + R(A,ω,Ψ),  2  0 0 −∇s +∆A   where R(A,ω,Ψ) is a differential operator of order one. In the case n = 2 it ∗ follows that the entry dAdA appearing in the above matrix is equal to ∆A. 2 2 Then all the three components of the leading order term of −∇s + B(A,ω,Ψ) are Laplacians on one of the bundles Ω1(I ×Y, ad(I ×P )) or Ω0(I ×Y, ad(I × P )), showing that this operator is elliptic. This fails to be the case if n ≥ 3 and hence the claim follows. To overcome the lack of ellipticity of its linearization in dimension n ≥ 3 we introduce the following modification of Eq. (3). This modification arises as the formal L2-gradient flow equation associated with the restriction of the functional J to the infinite-dimensional manifold (with singularities) ∗ X1 := {(A, ω) ∈ X | dAω = 0}, where we let X := A(P ) × Ωn−2(Y, ad(P )). Note that in dimension n = 2 it follows that X1 = X, and the system (3) studied so far appears as a special case of the restricted flow equations which we shall introduce next. The tangent space of X1 at (A, ω) ∈ X1 is T(A,ω)X1 = ker L(A,ω), where we define 1 n−2 n−3 L(A,ω) :Ω (Y, ad(P )) ⊕ Ω (Y, ad(P )) → Ω (Y, ad(P )), ∗ (α, v) 7→ dAv + ∗[∗ω ∧ α].

The formal adjoint of L(A,ω) is the operator

∗ n−3 1 n−2 L(A,ω) :Ω (Y, ad(P )) → Ω (Y, ad(P )) ⊕ Ω (Y, ad(P )), n+1 Ψ 7→ ((−1) ∗ [∗ω ∧ Ψ], dAΨ). ∗ Let (A, ω) ∈ X1. Because the condition dAω = 0 is gauge-invariant it follows that the image of the map 0 1 n−2 d(A,ω) :Ω (Y, ad(P )) → Ω (Y, ad(P )) ⊕ Ω (Y, ad(P )), ϕ 7→ (dAϕ, [ω ∧ ϕ]) is contained in ker L(A,ω). We therefore obtain an elliptic complex

d(A,ω) L(A,ω) 0 −→ Ω0(Y, ad(P )) −→ Ω1(Y, ad(P )) ⊕ Ωn−2(Y, ad(P )) −→ Ωn−3(Y, ad(P )) −→ 0 36 R.JANNERANDJ.SWOBODA

∗ ∗ together with a Laplace operator D(A,ω) := d(A,ω)d(A,ω) + L(A,ω)L(A,ω). Any ξ = (α, v) ∈ Ω1(Y, ad(P )) ⊕ Ωn−2(Y, ad(P )) admits a unique L2-orthogonal decomposition

ξ = ξ0 + ξ1 + d(A,ω)ϕ, ∗ 0 2 where ξ0 ∈ ker D(A,ω), ξ1 ∈ im L(A,ω), and ϕ ∈ Ω (Y, ad(P )). The L - orthogonal projection π(A,ω) : T(A,ω)X → T(A,ω)X1 is therefore given by

π(A,ω)ξ = ξ − ξ1, where ξ1 is the unique solution of the elliptic equation ∗ D(A,ω)ξ1 = L(A,ω)L(A,ω)ξ. n If in particular ξ = ((−1) ∗ dAω,ω − ∗FA), which is the negative of the L2-gradient of J , then it follows by a short calculation that ∗ n L(A,ω)L(A,ω)ξ = (− ∗ [∗ω ∧ ∗[∗ω ∧ ∗dAω]], (−1) dA ∗ [∗ω ∧ ∗dAω]).

In this case, ξ1 = (α1, v1) is the solution of the equation n (48) D(A,ω)ξ1 = (− ∗ [∗ω ∧ ∗[∗ω ∧ ∗dAω]], (−1) dA ∗ [∗ω ∧ ∗dAω]). Deﬁnition 23. Let I be an interval. The restricted L2-gradient ﬂow is the system of equations

0= ∂ A − d Ψ + (−1)n+1 ∗ d ω − α (49) s A A 1 (0= ∂sω + [Ψ,ω] − ω + ∗FA − v1 ∞ 0 for a triple (A, ω, Ψ) ∈ C (I, X1 × Ω (Y, ad(P ))). Here the correction term ξ1 = (α1, v1) is defined to be the solution of Eq. (48). The term Ψ is included in order to make (49) invariant under time-dependent gauge transformations. We conclude this section with a number of remarks. First one should ∗ note that the additional condition dAω = 0 imposed on pairs (A, ω) ∈ X is compatible with the critical point equation (4). Namely then, ω = ∗FA ∗ and hence dAω = 0 holds automatically by the Bianchi identity. Second, the n+1 term ((−1) ∗dAω−α1, −ω+∗FA−v1) appearing in (49) can be understood as the Riemannian gradient of the functional J : X1 → R, where we view 2 X1 as submanifold of X endowed with the L submanifold metric. Finally, the linearization of (49) together with a gauge-fixing condition as discussed in Remark 3 now leads to an elliptic system in any dimension n (in contrast to the unrestricted flow (3), cf. Proposition 22). A slight complication now comes from the fact that the term ξ1 = (α1, v1) in (49) is nonlocal (however of order zero). Still it is conceivable that an analysis of the moduli spaces of solutions of (49) is possible in analogy to that of the solutions of (3). This will lead to an elliptic Yang–Mills homology for manifolds of dimension n ≥ 3. We leave this programe to be carried out in a future publication. ELLIPTIC YANG–MILLS FLOW THEORY 37

Appendix A. Differential inequalities for the energy density Recall from §4 the gauge-invariant energy density of the solution (A, ω, Ψ) of (3), which we defined to to be the function 1 e(A, ω, Ψ) := (| ∗ d ω + d Ψ+ X (A, ω)|2 2 A A f 2 + | ∗ FA − ω + [Ψ,ω]+ Yf (A, ω)| ): I × Σ → R. Our aim is to derive an elliptic differential inequality satisfied by e(A, ω, Ψ). Let ∆Σ denote the (positive semidefinite) Laplace–Beltrami operator on (Σ, g), and define d2 ∆ := − +∆ . I×Σ ds2 Σ Proposition 24. Assume that (A, ω, Ψ) is a solution of (32). Then the energy density e = e(A, ω, Ψ) satisfies on I × Σ the differential inequality 6 3 (50) ∆I×Σe ≤ A0 + A1|ω| + A2e + A3e 2 for positive constants A0, A1, A2, A3 which do not depend on (A, ω, Ψ). 3 Remark 25. Note that the exponent 2 appearing in the differential inequality (52) is critical in dimension dim Y = 3 but subcritical in the present case. Proof. Since the asserted inequality is invariant under gauge transformations we may assume that Ψ = 0. We first show the claim in the unperturbed case (Xf ,Yf ) = 0. By differentiating (32) with respect to s we obtain A¨ = d∗ F + ∗d ω + ∗[A˙ ∧ ω] (51) A A A ω¨ = d∗ d ω +ω ˙ A A It then follows that

∆I×Σe d d = −| d ω|2 − | (−ω + ∗F )|2 − |∇ d ω|2 − |∇ (−ω + ∗F )|2 ds A ds A A A A A d2 d2 + hd ω, (− + ∇∗ ∇ )d ωi + h−ω + ∗F , (− + ∇∗ ∇ )(−ω + ∗F )i A ds2 A A A A ds2 A A A d d = −| d ω|2 − | (−ω + ∗F )|2 − |∇ d ω|2 − |∇ (−ω + ∗F )|2 ds A ds A A A A A d2 + hd ω, (− +∆ )d ω + {F , d ω} + {R , d ω}i A ds2 A A A A Σ A I d2 + h−| ω + ∗F , (− +∆ {z)(−ω + ∗F )+ {F , −ω +} ∗F } + {R , −ω + ∗F }i . A ds2 A A A A Σ A II In the last step we replaced ∇∗ ∇ using the standard Weizenböck formula | A A {z } ∗ ∇A∇A =∆A + {FA, · } + {RΣ, · }, 38 R.JANNERANDJ.SWOBODA where the brackets denote bilinear terms with fixed coefficients, and RΣ is some curvature expression determined by the fixed Riemannian metric g. It remains to analyze terms I and II. For the first one we obtain, making use of Eq. (51), d2 (− +∆ )d ω ds2 A A d = − (d ω˙ + [A˙ ∧ ω]) + d d∗ d ω + d∗ d d ω ds A A A A A A A ˙ ¨ ˙ ∗ ∗ = −dAω¨ − [A ∧ ω˙ ] − [A ∧ ω] − [A ∧ ω˙ ]+ dAdAdAω + dA[FA ∧ ω] d = −d ω˙ + d ∗ F − 2[A˙ ∧ ω˙ ] − [A¨ ∧ ω]+ d d∗ d ω + d∗ [F ∧ ω] A A ds A A A A A A ˙ ˙ ¨ ∗ ∗ = −dAω˙ + dA ∗ dAA − 2[A ∧ ω˙ ] − [A ∧ ω]+ dAdAdAω + dA[FA ∧ ω] ˙ ¨ ∗ = −dAω˙ − 2[A ∧ ω˙ ] − [A ∧ ω]+ dA[FA ∧ ω]

= −dAω˙ − 2[A˙ ∧ ω˙ ] − [(∗dAω + ∗[A˙ ∧ ω]) ∧ ω] − [∗FA ∧ ∗dAω]. By (51) again, term II reduces to d2 d (− +∆ )(−ω + ∗F ) =ω ¨ − ∗ d A˙ − ∆ ω + d∗ d ∗ F ds2 A A ds A A A A A ¨ ˙ ˙ ∗ =ω ¨ − ∗dAA − ∗[A ∧ A] − ∆Aω + dAdA ∗ FA ∗ ˙ ˙ ˙ =∆Aω +ω ˙ + dA[A ∧ ω] − ∗[A ∧ A]. We then estimate (for any ε> 0 and some absolute constant C > 0), using H¨older’s inequality several times

|I| ≤|hdAω, {FA, dAω} + {RΣ, dAω}i|

+ |hdAω, −dAω˙ − 2[A˙ ∧ ω˙ ] − [(∗dAω + ∗[A˙ ∧ ω]) ∧ ω] − [∗FA ∧ ∗dAω]i| 3 3 2 −1 2 3 3 3 6 ≤C(|dAω| + |FA| + ε|dAω˙ | + ε |dAω| + |A˙| + |ω˙ | + |ω| + |ω| ) and

|II| ≤|h−ω + ∗FA, {FA, −ω + ∗FA} + {RΣ, −ω + ∗FA}i| ∗ ˙ ˙ ˙ + |h−ω + ∗FA, ∆Aω +ω ˙ + dA[A ∧ ω] − ∗[A ∧ A]i| 3 3 2 −1 2 2 ≤C(|− ω + ∗FA| + |FA| + ε|∇AdAω| + ε |− ω + ∗FA| + |ω˙ | 2 −1 3 −1 6 3 3 + ε|∇AA˙| + ε |− ω + ∗FA| + ε |ω| + |∇Aω| + |A˙| ). Now we ﬁx ε 0, and δ > 0 such that the following holds for all metrics g ELLIPTIC YANG–MILLS FLOW THEORY 39

n n

R ½ R on such that kg − kW 1,∞ < δ. Let Br(0) ⊆ be the geodesic ball of ra- 2 dius 0 < r ≤ 1. Suppose that the nonnegative function e ∈ C (Br(0), [0, ∞)) satisﬁes for some A0, A1, a ≥ 0

(n+2)/n −n/2 ∆e ≤ A0 + A1e + ae and e ≤ µa . ZBr(0) Then

2 n/2 −n e(0) ≤ CA0r + C A1 + r e. ZBr(0) Proof. For a proof we refer to [28, Theorem 1.1].

This result together with Proposition 24 implies the following L∞ bound which we made use of in §5. Lemma 27. Assume that (A, ω, Ψ) is a solution of (32). Then the energy density e = e(A, ω, Ψ) satisﬁes on I × Σ the estimate

(52) kekL∞(I×Σ) ≤ CkekL1(I×Σ) for a positive constant C which only depends on |I| and on Σ. Proof. Theorem 26 along with Proposition 24 yields for suitable constants C0 and C1 and every (s, z) ∈ I × Σ the estimate

e(s, z) ≤ C0(kωkL∞(I×Σ) + kAkL∞(I×Σ))+ C1 e. ZI×Σ Taking the supremum over I × Σ the claim follows, since we may absorb the second term using for ε> 0 suﬃciently small the estimate −1 kωkL∞(I×Σ) ≤ εk∇AωkL∞(I×Σ) + ε kωkL2(I×Σ), and likewise for kAkL∞(I×Σ).

Appendix B. Further auxiliary results By gauge-invariance of the subsequent estimates we may in the following assume that Φ = 0. Let (A, ω) be a solution of (3) such that for critical points (A±,ω±) the asymptotic condition lim (A(s),ω(s)) = (A±,ω±) s→±∞ ± ± ± 2 is satisfied. Let C := (J + hf )(A ,ω ). Since (A, ω) is an L -gradient flow line of J + hf it satisfies the energy identity ∞ 2 − + (53) k∇(J + hf )(A(s),ω(s))kL2 (Σ) ds = C − C ≥ 0. Z−∞ 2 From this we obtain the following estimate for the L -norm of FA. 40 R.JANNERANDJ.SWOBODA

Lemma 28. For (A, ω) as above and any interval I ⊆ R there holds the estimate 1 2 − + kF k 2 ≤ (1 + |I|)C − C + |I|C 2 A L (I×Σ) f for a positive constant Cf which only depends on the perturbation hf . 2 2 Proof. Since for all (A, ω), k∇(J + hf )(A, ω)kL2(Σ) ≤ kω − ∗FA − Yf kL2(Σ) it follows from the energy identity (53) that

− + 2 C − C ≥ kω − ∗FA − Yf kL2(Σ) ds ZI 1 2 2 ≥ kω − ∗F k 2 − kY k 2 ds 2 A L (Σ) f L (Σ) ZI 1 2 2 2 = kF k 2 − 2hω, ∗F i + kωk 2 − kY k 2 ds 2 A L (Σ) A L (Σ) f L (Σ) ZI 1 2 2 = kF k 2 −J (A, ω) − kY k 2 ds 2 A L (Σ) f L (Σ) ZI 1 2 − ≥ kF k 2 − |I|C − |I|C . 2 A L (I×Σ) f For the last estimate we used that J (A(s),ω(s)) ≤ C− for all s ∈ I and 2 further that kYf kL2(Σ) ≤ Cf for a constant Cf which does not depend on (A, ω) (cf. [21, Proposition D.1 (iii)]). The claim hence follows. An estimate for the L2-norm of ω is obtained in the following lemma. Lemma 29. For (A, ω) as above and every s ∈ R there holds the estimate 1 kω(s)k 2 ≤ kF k 2 + C . 2 L (Σ) A(s) L (Σ) f for a positive constant Cf which only depends on the perturbation hf . + + + Proof. Note that ω = FA+ because (A ,ω ) ∈ Crit(J +hf ), and therefore

+ + + 1 2 + + C = (J + h )(A ,ω )= |F + | dvol +h (A ) ≥ h (A ). f 2 A Σ f f ZΣ It therefore follows from the gradient ﬂow property and the Cauchy–Schwartz inequality that for all s ∈ R + hf (A ) ≤ (J + hf )(A(s),ω(s)) 1 = hF ,ω(s)i− |ω(s)|2 dvol +h (A(s),ω(s)) A(s) 2 Σ f ZΣ 1 2 ≤ kF k 2 kω(s)k 2 − kω(s)k 2 + h (A(s),ω(s)). A(s) L (Σ) L (Σ) 2 L (Σ) f By deﬁnition of hf there exists a constant Cf such that |hf (A, ω)|≤ Cf for all (A, ω). Thus we can further estimate

1 2 kω(s)k 2 − kF k 2 kω(s)k 2 − 2C ≤ 0, 2 L (Σ) A(s) L (Σ) L (Σ) f ELLIPTIC YANG–MILLS FLOW THEORY 41 which implies the result.

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(R. Janner) Alpiq Management AG, Bahnhofquai 12, CH-4601 Olten, Switzer- land E-mail address: [email protected] URL: www.remijanner.ch

(J. Swoboda) Max-Planck-Institut fur¨ Mathematik, Vivatsgasse 7, D-53111 Bonn, Germany E-mail address: [email protected] URL: http://www.mpim-bonn.mpg.de/de/node/94